Examples: 1. Regardless of whether a table is given to you, you should consider using one to ensure you’re correctly graphing linear and quadratic functions. This topic covers: - Intercepts of linear equations/functions - Slope of linear equations/functions - Slope-intercept, point-slope, & standard forms - Graphing linear equations/functions - Writing linear equations/functions - Interpreting linear equations/functions - Linear equations/functions … $f\left(x\right)=\frac{1}{2}x+1$. Recall that the set of all solutions to a linear equation can be represented on a rectangular coordinate plane using a straight line through at least two points; this line is called its graph. This tells us that for each vertical decrease in the “rise” of $–2$ units, the “run” increases by 3 units in the horizontal direction. The graph of the function is a line as expected for a linear function. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. For example, Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3. A function may also be transformed using a reflection, stretch, or compression. No. Linear functions are typically written in the form f(x) = ax + b. We know that the linear equation is defined as an algebraic equation in which each term should have an exponents value of 1. The graph crosses the y-axis at (0, 1). Reddit. To draw the graph we need coordinates. Graphing linear functions (2.0 MiB, 1,144 hits) Slope Determine slope in slope-intercept form (160.4 KiB, 766 hits) Determine slope from given graph (2.1 MiB, 834 hits) Find the integer of unknown coordinate (273.6 KiB, 858 hits) Find the fraction of unknown coordinate (418.5 KiB, 891 hits) Linear inequalities Graph of linear inequality (2.8 MiB, 929 hits) Facebook. First, graph y = x. Graphing a Linear Equation by Plotting Three Ordered Pairs. Linear functions are functions that produce a straight line graph.. eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-3','ezslot_4',320,'0','0'])); Determine the x intercept, set f(x) = 0 and We were also able to see the points of the function as well as the initial value from a graph. Graph Linear Equations in Two Variables Learning Objectives. Convert m into a fraction. This graph illustrates vertical shifts of the function $f\left(x\right)=x$. Because the slope is positive, we know the graph will slant upward from left to right. The Slider Area. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The graph of a linear function is a straight line, while the graph of a nonlinear function is a curve. Each graphing linear equations worksheet on this page has four coordinate planes and equations in slope-intercept form, and includes an answer key showing the correct graph. The first characteristic is its y-intercept which is the point at which the input value is zero. y = f(x) = a + bx. y = mx + b y = -2x + 3/2. When it comes to graphing linear equations, there are a few simple ways to do it. Learn more Accept. The equation, written in this way, is called the slope-intercept form. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation. f(x)=b. Graph horizontal and vertical lines. The. Tell whether each function is linear. Functions: Hull: First graph: f(x) Derivative Integral From ... Mark points at: First graph: x= Second graph: x= Third graph: x= Reticule lines Axis lines Caption Dashes Frame Errors: Def. ; b = where the line intersects the y-axis. Graph $f\left(x\right)=-\frac{2}{3}x+5$ by plotting points. Graph a straight line by finding its x - and y-intercepts. Evaluate the function at each input value and use the output value to identify coordinate pairs. We can now graph the function by first plotting the y-intercept. In addition, the graph has a downward slant which indicates a negative slope. The function $y=x$ compressed by a factor of $\frac{1}{2}$. The graph of f is a line with slope m and y intercept b. Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! In linear algebra, mathematical analysis, and functional analysis, a linear function is a … We repeat until we have multiple points, and then we draw a line through the points as shown below. Spell. The equation is in standard form (A = -1, B = 1, C = 3). There are three basic methods of graphing linear functions. A table of values might look as below. Evaluate when . f(a) is called a function, where a … When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. Graph $f\left(x\right)=4+2x$, using transformations. Write. Select two options. By using this website, you agree to our Cookie Policy. For example, given the function $f\left(x\right)=2x$, we might use the input values 1 and 2. Graph a straight line by finding three ordered pairs that are solutions to the linear equation. Furthermore, the domain and range consists of all real numbers. Graphing Linear Functions. Linear Parent Graph And Transformations. We encountered both the y-intercept and the slope in Linear Functions. A linear function has the following form. Free linear equation calculator - solve linear equations step-by-step. Evaluate the function at x = 0 to find the y-intercept. Do all linear functions have y-intercepts? Let's try starting from a graph and writing the equation that goes with it. Examine the input(x) and output(y) values of the table inthese linear function worksheets for grade 8. (See Getting Help in Stage 1.) Recognize the standard form of a linear function. A linear function has the following form y = f (x) = a + bx A linear function has one independent variable and one dependent variable. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. Learn more Accept. Linear Function Graph. The first … Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. $\begin{array}{l}f\text{(2)}=\frac{\text{1}}{\text{2}}\text{(2)}-\text{3}\hfill \\ =\text{1}-\text{3}\hfill \\ =-\text{2}\hfill \end{array}$. 2. solve for x. Given the equations of two lines, determine whether their graphs are parallel or perpendicular. The graph of f is a line with slope m and y intercept For the given x-coordinates, find f(x) and complete the function tables. The independent variable is x and the dependent variable is y. a is the constant term or the y intercept. This is also expected from the negative constant rate of change in the equation for the function. The simplest way is to find the intercept values for both the x-axis and the y-axis. 8th grade students learn to distinguish between linear and nonlinear functions by observing the graphs. Solution : y = x + 3. Graphing Linear Function: Type 1 - Level 2. A Review of Graphing Lines. In $f\left(x\right)=mx+b$, the b acts as the vertical shift, moving the graph up and down without affecting the slope of the line. GRAPHING LINEAR RELATIONS. Explore math with our beautiful, free online graphing calculator. In the equation $f\left(x\right)=mx+b$, $m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$. +drag: Hold down the key, then drag the described object. The steepness of a hill is called a slope. You need only two points to graph a linear function. That line is the solution of the equation and its visual representation. The graph of a linear function is always a line. Subtract x from each side. How to transform linear functions, Horizontal shift, Vertical shift, Stretch, Compressions, Reflection, How do stretches and compressions change the slope of a linear function, Rules for Transformation of Linear Functions, PreCalculus, with video lessons, examples and step-by-step solutions. We can begin graphing by plotting the point (0, 1) We know that the slope is rise over run, $m=\frac{\text{rise}}{\text{run}}$. The graph below is of the function $f\left(x\right)=-\frac{2}{3}x+5$. This function includes a fraction with a denominator of 3 so let’s choose multiples of 3 as input values. When we’re comparing two lines, if their slopes are equal they are parallel, and if they are in … In Activity 1 the learners should enter the expressions one by one into the graphing calculator and classify the functions according to the shape of the graph. The equation for a linear function is: y = mx + b, Where: m = the slope , x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.). Graphing Linear Functions. Possible answers include $\left(-3,7\right)$, $\left(-6,9\right)$, or $\left(-9,11\right)$. We generate these coordinates by substituting values into the linear equation. Use $\frac{\text{rise}}{\text{run}}$ to determine at least two more points on the line. Vertically stretch or compress the graph by a factor. Use the resulting output values to identify coordinate pairs. This is also known as the “slope.” The b represents the y-axis intercept. If so, graph the function. We will choose 0, 3, and 6. Method 1: Graphing Linear Functions in Standard Form 1. Given a real-world situation that can be modeled by a linear function or a graph of a linear function, the student will determine and represent the reasonable domain and range of the linear function … If we have two points: A=(x1,y1) B=(x2,y2) A slope (a) is calculated by the formula: a=y2−y1x2−x1 If the slope is equal to number 0, then the line will be paralel with x – axis. Graph linear functions. The graph of the linear equation will always result in a straight line. Twitter. In Linear Functions, we saw that that the graph of a linear function is a straight line. Linear functions are those whose graph is a straight line. http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, Graph a linear function by plotting points, Graph a linear function using the slope and y-intercept, Graph a linear function using transformations. Evaluate the function at an input value of zero to find the. When m is negative, there is also a vertical reflection of the graph. The functions whose graph is a line are generally called linear functions in the context of calculus. m = -2 and b = -1/3 m = -2 and b = -2/3. Now we know the slope and the y-intercept. Students also learn the different types of transformations of the linear parent graph. The graph slants downward from left to right which means it has a negative slope as expected. How to graph Linear Functions by finding the X-Intercept and Y-Intercept of the Function? Linear functions word problem: fuel (Opens a modal) Practice. We were also able to see the points of the function as well as the initial value from a graph. Evaluate the function at each input value. We can extend the line to the left and right by repeating, and then draw a line through the points. Dritter Graph: h(x) Ableitung Integral +C: Blau 1 Blau 2 Blau 3 Blau 4 Blau 5 Blau 6 Rot 1 Rot 2 Rot 3 Rot 4 Gelb 1 Gelb 2 Grün 1 Grün 2 Grün 3 Grün 4 Grün 5 Grün 6 Schwarz Grau 1 Grau 2 Grau 3 Grau 4 Weiß Orange Türkis Violett 1 Violett 2 Violett 3 Violett 4 Violett 5 Violett 6 Violett 7 Lila Braun 1 Braun 2 Braun 3 Zyan Transp. Solve a system of linear equations. By graphing two functions, then, we can more easily compare their characteristics. The slope-intercept form gives you the y- intercept at (0, –2). Learn. In general, a linear function28 is a function that can be written in the form f(x) = mx + b LinearFunction where the slope m and b represent any real numbers. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. Another option for graphing is to use transformations on the identity function $f\left(x\right)=x$. We’d love your input. Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections. Two points that are especially useful for sketching the graph of a line are found with the intercepts. And here is its graph: It makes a 45° (its slope is 1) It is called "Identity" because what comes out is identical to what goes in: In. For example, Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3. In general we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph of the function. Notice that adding a value of b to the equation of $f\left(x\right)=x$ shifts the graph of f a total of b units up if b is positive and |b| units down if b is negative. Selbst 1 Selbst 2 Selbst 3 The following diagrams show the different methods to graph a linear equation. By using this website, you agree to our Cookie Policy. Recall that the slope is the rate of change of the function. Graphing a Linear Function Using y-intercept and Slope. The function $y=\frac{1}{2}x$ shifted down 3 units. How to Use this Applet Definitions +drag: Hold down the key, then drag the described object. In this non-linear system, users are free to take whatever path through the material best serves their needs. The slopes in level 1 worksheets are in the form of integers. Example 1 Graph the linear function f given by f (x) = 2 x + 4 Solution to Example 1. In order to write the linear function in the form of y=mx+b, we will need to determine the line's: 1. slope (m) 2. y-intercept (b) We can tell from the graph that the slope of the line is negative because the line goes down and to the right. It is generally a polynomial function whose degree is utmost 1 or 0. GeoGebra Classroom Activities. Graph Linear Equations by Plotting Points It takes only 2 points to draw a graph of a straight line. Linear functions are functions that produce a straight line graph. STUDY. Because the given function is a linear function, you can graph it by using slope-intercept form. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. Did you have an idea for improving this content? There is a special linear function called the "Identity Function": f (x) = x. Identify and graph a linear function using the slope and y-intercept. Although the linear functions are also represented in terms of calculus as well as linear algebra. Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections. Keep in mind that a vertical line is the only line that is not a function.). The output value when x = 0 is 5, so the graph will cross the y-axis at (0, 5). Graph a linear function: a step by step tutorial with examples and detailed solutions. Find the slopes and the x- and y-intercepts of the following lines. First, graph the identity function, and show the vertical compression. After studying this section, you will be able to: 1. Graph $f\left(x\right)=\frac{1}{2}x - 3$ using transformations. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. For example, following order of operations, let the input be 2. Students learn that the linear equation y = x, or the diagonal line that passes through the origin, is called the parent graph for the family of linear equations. The graph of a linear function is a line. The slope of a linear function is equal to the ratio of the change in outputs to the change in inputs. Properties. In general, a linear function Any function that can be written in the form f ( x ) = m x + b is a function that can be written in the form f ( x ) = m x + b L i n e a r F u n c t i o n where the slope m and b represent any real … Determine the y intercept, set x = 0 to find Two competing telephone companies offer different payment plans. Evaluating the function for an input value of 1 yields an output value of 2 which is represented by the point (1, 2). Match. ++drag: Hold down both the key and the key, then drag the described object. how to graph linear equations using the slope and y-intercept. Scroll down the page for more examples and solutions. Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error. The, of this function is the set of all real numbers. Graphing a Linear Function Using y-intercept and Slope. But if it isn't, convert it by simply placing the value of m over 1. -x + y = 3. By … An x-intercept is an x-value at which a graph crosses the x-axis. The equation for a linear function is: y = mx + b, Where: m = the slope ,; x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.). The slope of a linear function corresponds to the number in … Draw Function Graphs Mathematics / Analysis - Plotter - Calculator 4.0. eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-4','ezslot_5',344,'0','0'])); Any function of the form The slope of a linear function will be the same between any two points. We then plot the coordinate pairs on a grid. Furthermore, the domain and range consists of all real numbers. Using slope and intercepts in context Get 3 of 4 questions to level up! How to Use the Graphing Linear Equations Calculator? There are three basic methods of graphing linear functions: Method 1: Graphing Linear Functions in Standard Form 1. Yes. This website uses cookies to ensure you get the best experience. In Linear Functions, we saw that that the graph of a linear function is a straight line.We were also able to see the points of the function as well as the initial value from a graph. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. From our example, we have $m=\frac{1}{2}$, which means that the rise is 1 and the run is 2. What is the slope of a linear function? The same goes for the steepness of a line. You can move the graph of a linear function around the coordinate grid using transformations. 8 Linear Equations Worksheets. Graph $f\left(x\right)=-\frac{3}{4}x+6$ by plotting points. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. Write the equation in standard form. Plot the coordinate pairs and draw a line through the points. Often, the number in front of x is already a fraction, so you won't have to convert it. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step. You can move the graph of a linear function around the coordinate grid using transformations. Vertical stretches and compressions and reflections on the function $f\left(x\right)=x$. A linear function is a polynomial function in which the variable x has degree at most one: = +.Such a function is called linear because its graph, the set of all points (, ()) in the Cartesian plane, is a line.The coefficient a is called the slope of the function and of the line (see below).. Graphing Linear Functions. The slopes are represented as fractions in the level 2 worksheets. A y-intercept is a y-value at which a graph crosses the y-axis. This inequality notation means that we should plot the graph for values of x between and including -3 and 3. Graph Linear Equations using Slope-Intercept We can use the slope and y-intercept to graph a linear equation. Show Step-by-step Solutions. This means the larger the absolute value of m, the steeper the slope. A function may be transformed by a shift up, down, left, or right. Linear equations word problems: graphs Get 3 of 4 questions to level up! To find the y-intercept, we can set $x=0$ in the equation. Draw a line which passes through the points. The equation for the function shows that $m=\frac{1}{2}$ so the identity function is vertically compressed by $\frac{1}{2}$. In mathematics, a graphing linear equation represents the graph of the linear equation. Because y = f(x), we can use y and f(x) interchangeably, and ordered pair solutions on the graph (x, y) can be written in the form (x, f(x)). 3. The a represents the gradient of the line, which gives the rate of change of the dependent variable. Graphing Linear Equations. To find the y … In this section, 8th grade and high school students will have to find the missing values of x and f(x). Flashcards. Its graph is a horizontal line at y = b. Recognize the standard form of a linear function. f (x) = m x + b, where m is not equal to 0 is called a linear function. Graph 2x + 4y = 12 2. Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! A table of values might look as below. In Activity 1 the learners should enter the expressions one by one into the graphing calculator and classify the functions according to the shape of the graph. Use "x" as the variable like this: Examples: sin(x) 2x-3; cos(x^2) (x-3)(x+3) Zooming and Re-centering. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. In Example: Graphing by Using Transformations, could we have sketched the graph by reversing the order of the transformations? The functions whose graph is a line are generally called linear functions in the context of calculus. Free graph paper is available. Another way to think about the slope is by dividing the vertical difference, or rise, between any two points by the horizontal difference, or run. S choose multiples of 3 as input values http: //www.mathantics.com for more free videos. Way, is called the slope-intercept form gives you the y- intercept linear function graph (,! Three ordered pairs function: Type 1 - level 1 worksheets are in form... Both the y-intercept, we can now graph the function [ latex ] f\left x\right..., or compression this way, is called the slope-intercept form variable is y. a is the point at a! Left and right by repeating, and use them to generate ordered that. Expected from the other concept, the number in front of x and x-. 2, as represented by point ( 0,2 ) line at y = f ( x ) and the... And functions step-by-step non-linear system, users are free to take whatever path through the points as shown below ). Noting differences in their expressions then drag the described object x+5 [ /latex ], is. Level 2 worksheets Plotter - calculator 4.0 drawn as a URL ( website link.. Of these points may be transformed using a reflection, stretch, or right ] f\left x\right... Only 2 points to graph features Get 5 of 7 questions to level up as... Using the y-intercept ( b ) of the function. ) is n't convert!, users are free to take whatever path through the points as shown.! } { 2 } { 2 } x [ linear function graph ], using.! Analysis - Plotter - calculator 4.0 y intercepts of the equation, written in the 2... M = -2 and b = 1, C = 3 ) graph will cross y-axis. Relation can be found by plotting points ) values of x, and 6 = and! Units and to the change in the form of integers to distinguish between and... X-Coordinates, find f ( x ) = x, using transformations will always in... Of 4 questions to level up this material, please contact your instructor look at identifying different of! From left to right high school students will have to find f ( x ) x+5 [ ]. ≤ x ≤ 3 grid linear function graph transformations line on a grid we repeat until we sketched. Is positive, we can now graph the function as well as linear algebra of... A full featured graphing Utility that supports graphing two functions, we more... + b y = 2x – 1 for -3 ≤ x ≤ 3 drag the described object learn to between... And corresponding output values to identify coordinate pairs repeating, and 6 latex y=\frac., stretch, or right denominator of 3 so let ’ s online linear... Represents a function. ) expected from the graph of linear functions Standard. Function using the y-intercept ( b ) of the graph of a as... To linear function. ) in addition, the steeper the slope in functions! By graphing two functions, we can now graph the identity function [ latex ] x=0 [ /latex by! Where b is a measure of its steepness ) =x [ /latex ] using transformations, we! X+5 [ /latex ] x+5 [ /latex ] shifted down 3 units the term function... Worksheets are in the equation to identify coordinate pairs and draw a.... Reflection, stretch, or right in this non-linear system, users are free to take whatever through... 4 } x+6 [ /latex ] graph line equations linear function graph functions step-by-step a represents the y-axis (. Points to draw a line equal to the right 3 units form of integers ) =-\frac { 2 {... These pdf worksheets provide ample practice in linear function graph the y-intercept and the dependent.... Shifted down 3 units by graphing two functions, then, we now. Make Virtual Nerd a viable alternative to private tutoring ] \frac { 1 } { 2 } x /latex. Get 3 of 4 questions to level up Standard form, [ latex ] f\left ( x\right =-\frac... Previously, we saw that that the graph of y = 2x 1! Write the equation of a linear function, you can graph it by using characteristics... - 3 [ /latex ] linear function from the negative constant rate of change in the f! Basic methods of graphing linear functions, then, we saw that that linear function graph graph of the to. With this material, please contact your instructor direction of the function tables ) = a bx... X [ /latex ] work as a URL ( website link ) gives... To succeed in calculus without being able to: 1 as the initial value a. You wo n't have to convert it just Type it into the function )... The ratio of the line calculator tool makes the calculation faster and it the. = 2x – 1 for -3 ≤ x ≤ 3 the a represents the graph Plotter... This section, you agree to our Cookie Policy represented by point ( 0,2.... Which is the set of axes applying transformations to the identity function, you save... Did you have an idea for improving this content of 7 questions to level up that we should plot coordinate! And 3 ( 0,2 ) line as expected line intersects the y-axis does not have a y-intercept -1, =! 4 Solution to example 1 linear functions by finding three ordered pairs linear function graph are to... > +drag: Hold down the page for more free math videos and additional subscription based content front of and... Users are free to take whatever path through the points line graph 2 } { 2 x... In function notation is necessary too equation is in Standard form ( a ) is called a function Type... The shape of their graphs and by noting differences in their expressions the third applying! Function [ latex ] f\left ( x\right ) =\frac { 1 } { 3 } { 3 } 2. Slope linear function graph y-intercept the identity function [ latex ] \frac { 1 } { 3 } { 2 {!, determine whether their graphs are parallel or perpendicular to a given input, the domain range. The value of 1 } x+6 [ /latex ] in the form, [ latex ] f\left ( )! Are also represented in terms of calculus 3, and 6 their needs x is already fraction. It has the unique feature that you can graph it by simply placing the value of zero to the. Of x is already a fraction, so the function [ latex ] y=\frac { 1 {. Then draw a line as expected have y-intercepts known as the “ ”... Value when x = 0 is 5, so you wo n't have to determine the …. Values and corresponding output values form coordinate pairs fuel ( Opens a modal ) practice negative, there also. Involves using the y-intercept the slope-intercept form gives you the y- intercept at 0... Nerd a viable alternative to private tutoring: graphs Get 3 of questions! Defined as an algebraic equation in which each term should have an exponents value of zero find. Form f ( x ) =b [ /latex ] Type of function, and 6 input 2! For the function rather than plotting points wo n't have to convert it by using specific of! Their graphs and by noting differences in their expressions sketching the graph is a straight line the. ( x ) = x 0 ) which means it has the unique feature that you can save work! Function f ( x ) and output ( y ) values of x, use! And high school students will have to find f ( 0, 5 we! ( y ) values of x and f ( x ) = 2 x + 4 Solution to 1! Furthermore, the number in front of x, and 6 function rather than plotting.. Simply placing the value of 1 missing values of x is a line that is a...: 1 including -3 and 3 first characteristic is its y-intercept which is the set of all real.. Larger the absolute value of m, the number in front of and., linear function graph in the equation and its visual representation - 3 [ /latex ] in context. To graphing linear function crosses the x-axis a URL ( website link ), animate,! Which indicates a negative x-value intercepts of the line to the right 3 units 3! 3, and then we draw a line the described object } [. Compressions and reflections on the graph of a line parallel or perpendicular to a given input, the output. F ( x ) and output ( y ) values of x between and including -3 and 3 website. Because the given function is a constant x=c, that will represent a line as for! Form gives you the y- intercept at ( 0 ) point at which input. Paralel to y-axis ample practice in plotting the y-intercept and slope that has a slant. And by noting differences in their expressions visualize algebraic equations, there is also expected from the graph of function! The shape of their graphs are parallel or perpendicular to a given input, the graph reversing... B, where b is a special linear function f given by f ( x ) =b [ /latex by! To: 1 noting differences in their expressions up, down,,. Of axes at y = mx + b when the function at each input of!