Graphs, Linear Equations and Functions There are several ways to graph a linear equation: Make a table of values Use slope and y-intercept Use x and y intercepts Oct 5 9:37 PM Oct 5 9:38 PM Example: Make a table of values and graph 2x+y=5 x 0 1 2 y 1 Find the missing value in the table for 2x y = 10. A 7 B 2 x y 1 12 5 0 4? C 7 D 2 Oct 5 9:43 PM Oct 6 4:20 PM 2 Find the value of y when x = 6 3 Point A lies on the line x + 3y = 7. Find the missing coordinate of point A(, 3). Oct 6 4:38 PM Oct 6 4:42 PM 1
4 True or False: The line 2x 6y = 6 goes through the origin. True False Using x and y Intercepts Example: Graph 3x+2y=6. Find the value of x when y=0. This is where the line crosses the x-axis, and is called the x-intercept. Substituting 0 for y, the equation becomes 3x=6, or x=2. The coordinates of the x-intercept are (2, 0). Now find the value of y when x=0. This is where the line crosses the y-axis, called the y-intercept. Substituting 0 for x, the equation becomes 2y=6, or y=3. The coordinates of the y-intercept are (0,3). Oct 6 4:51 PM Oct 5 9:55 PM Graph 3x+2y=6 Using the two intercepts, (2,0) and (0,3), we then graph the line. Remember: The x-intercept is where the line crosses the x-axis. The y-coordinate is zero on the x axis. y=0 x=0 The y-intercept is where the line crosses the y-axis. The x-coordinate is zero on the y-axis. x=0 Oct 5 10:03 PM Oct 6 4:20 PM Vertical lines are parallel to the y-axis. These lines have no y-intercept and have an equation in the form x=k, where k is a constant.for example, the vertical line x=2 is parallel to the y-axis and has an x-intercept of (2,0). Horizontal lines are parallel to the x-axis, so they have no x-intercept. The equations of horizontal lines are in the form y=k, where k is a constant. For example, y=-5 is parallel to the x-axis and has a y- intercept of (0,-5). Oct 6 5:03 PM Oct 6 5:41 PM 2
5 What is the y intercept of? 6 What is the x intercept of? Oct 6 5:51 PM Oct 6 5:53 PM 7 Which is the equation of a horizontal line? A B 8 Which is the graph of? A B C D C D Oct 6 5:56 PM Oct 6 6:01 PM 9 Which line goes through the origin? A B C Slope-Intercept Form The equation for a line may be written as y = mx + b, where m is the slope of the line and b is the y-intercept. For example, the line y=3x+1 has a slope of 3 and a y-intercept of (0,1). Recall that the slope of a line is found by measuring. A line which climbs from left D to right has a positive slope and a line which falls from left to right has a negative slope. Oct 6 6:10 PM Oct 6 6:15 PM 3
Unit 3 Linear Functions pt 1.notebook 10 What is the slope of this line? 11 What is the slope of this line? Oct 6 6:24 PM 12 What is the slope of this line? Oct 6 6:29 PM Graphing Linear Inequalities Linear inequalities are shaded regions of the coordinate plane with boundaries that are straight lines When we need to graph an inequality such as 2x+3y 6, we see that the boundary line has the equation 2x+3y=6. Once we graph the boundary line, we find the region to be shaded by testing a point on either side of the line. If the point makes the inequality true, it is part of the region to be shaded. If it does not make the inequality true, then the region on the opposite side of the boundary line must be shaded. Oct 6 6:33 PM Graphing 2x+3y 6: Graph the boundary line 2x+3y=6. Since the inequality allows it to be equal to 6, we use a solid line. (Otherwise, the line is dotted). The x-intercept is 3 and the yintercept is 2. So, the line passes through (3,0) and (0,2) Now test a point on one side of the line. We can use (0,0) as long as the line does not pass through the origin. Is 2(0)+3(0) 6? Yes, 0 6. So, we shade the side of the line that contains (0,0). Oct 6 7:52 PM Oct 6 7:38 PM Try this one: 4x-y 8 Oct 6 8:02 PM 4
Unit 3 Linear Functions pt 1.notebook Finding the Slope of a Line Does your graph look like this? When the equation of the line is in the form Ax+By=C where A, B and C are integers and A is positive, it is in standard form. We can find the slope of the line in standard form by solving the equation for y. Example: 5x-2y=8 Subtract 5x from both sides -2y=-5x+8 Divide both sides by -2 y=(5/2)x-4 From y=mx+b, we can see that the slope is 5/2 and the y-intercept is (0,-4). Oct 6 8:04 PM Oct 6 6:34 PM 14 What is the y intercept of the line 13 Find the slope of the line? Oct 6 6:55 PM Oct 6 6:57 PM 15 Find the slope of the line which goes through (3, 1) and ( 2, 4). Finding the slope from 2 points on the line: The slope formula can be used to find the slope of a line from the coordinates of any two points on the line. Example: The slope of the line which goes through (1,3) and (5,9) is equal to Oct 6 6:59 PM Oct 6 7:11 PM 5
Note! It is impossible to find the slope of a vertical line because the value of x is constant on the line. We say that the slope of a vertical line is undefined. If we try to use the slope formula, the denominator becomes zero, and it is impossible to divide by zero. Point-Slope Form: This form of an equation for a straight line is based on the fact that the slope of the straight line is the same between any two points on the line. Using point-slope form is an easy way to find the equation for a line when we know the slope, m, and one of the points it passes through (x 1, y 1 ). Oct 6 7:03 PM Oct 6 7:17 PM Example: Let's find an equation for the line that has a slope of 2 and passes through (3, 5). Using point-slope form, Substitute the values for m, x 1 and y 1. Use the distributive property. Add 5 to both sides. Now that the equation is in slope-intercept form, we can see that the y-intercept is (0,-1). Summary: Forms of Linear Equations Equation Description When to Use Slope-intercept form. Slope is m, y-intercept is (0,b). Graphing the equation using the y-intercept and rise/run from the slope. Point slope form. Slope is m and the line passes through (x 1, y 1) Finding the equation of a line when the slope and one or two points on the line are known Standard Form. A, B and C are integers and A is positive. Slope is - The x and y intercepts can be found quickly and used to graph A/B, B 0. x-intercept is (C/A, 0), C 0, the equation without having to y-intercept is (0,C/B), B 0. calculate the slope. If the graph intersects only the Horizontal Line. Slope is 0, and y- y axis, then y is the only variable intercept is (0,b). in the equation. If the graph intersects only the Vertical Line. Slope is undefined. x- x axis, then x is the only variable intercept is (a, 0). in the equation. Oct 6 7:24 PM Oct 6 9:12 PM If we know the slope of a line and the y- intercept, we can just substitute into slopeintercept form. Example: Write an equation for the line with slolpe of -2 and y-intercept of (0,7). Using y=mx+b, we get y=-2x+7 for the line. If we know the slope of the line and a point is passes through, we can use point-slope form. Example: Find an equation for the line that passes through the point (3, 7) as is parallel to y=3x-1. We know the slope has to be 3, because parallel lines have the same slope. Using point-slope form, we get Oct 7 2:30 PM Oct 7 2:34 PM 6
Write an equation for a line that is perpendicular to y=2x-3 and passes through the point (1,5). The slope of the line has to be the negative reciprocal of 2 to make it perpendicular to a line with slope 2. Again,using point-slope form, we get If we need to find an equation for the line parallel to a line in standard form, there is an easier way. If you notice, parallel lines when written in standard form have the same coefficients for x and y. Example: Find an equation for the line parallel to 2x+3y=9 that passes through (-3, 4). We know the line will be 2x+3y=C, and we can substitute for x and y from the point to find C. 2(-3)+3(4) = -6+12=6, so C=6, and the equation of the line is 2x+3y=6 Oct 7 2:41 PM Oct 7 2:47 PM If we need to find an equation for the line perpendicular to a line in standard form, there is an easier way. If you notice, perpendicular lines when written in standard form have the coefficients for x and y switched and the opposite sign between the two terms. Example: Find an equation for the line perpendicular to 2x+3y=9 that passes through (1,3). We know the line will be 3x-2y=C, and we can substitute for x and y from the point to find C. 3(1)-2(3) = 3-6=-3, so C=-3, and the equation of the line is 3x-2y=-3 Write an equation for the line parallel to y=3x 1 which goes through the point (2,4). Oct 7 2:54 PM Oct 8 5:09 PM Write an equation for the vertical line which passes through ( 3, 2). Write an equation for the line which is perpendicular to 3x 4y=16 and passes through ( 3,4). Oct 8 5:10 PM Oct 8 5:11 PM 7
Write an equation for the line which is perpendicular to y=1/2x+3 and goes through (0,1). Write an equation for the line with a slope of 5 which passes through (0,3). Oct 8 5:13 PM Oct 8 5:14 PM Write an equation for the line with a slope of 4 which passes through the point (1/2, 3). The table below shows the coordinates of points on a line. What is an equation for the line? x y 5 7 7 11 10 17 Oct 8 5:15 PM Oct 8 5:15 PM Applications of Linear Equations Suppose you are filling your car with gasoline. The price at the local gas station is $3.60 per gallon. Write an equation that describes the cost y to buy x gallons of gas. As you pump the gas, two numbers spin by, the number of gallons purchased and the price. The table summarizes the ordered pairs (x,y) Number of Gallons Purchased=x Price of the Number of Gallons=y 0 0(3.60)=$0 1 1(3.60)=$3.60 2 2(3.60)=$7.20 3 3(3.60)=$10.80 4 4(3.60)=$14.40 The equation y = 3.6x models this relationship, since we multiply the number of gallons by $3.60 to get the price. Suppose the same gas station was offering a car wash for $3.00 with every fill up. Write an equation to find the price for gas and a car wash. Since an additional $3.00 be charged, the equation becomes y= 3.6 x + 3 What would the price be for 7 gallons of gas and a car wash? $28.20 Oct 6 9:14 PM Oct 7 10:50 AM 8
Linear Models Sometimes data collected over a period time can be modelled by an equation for a straight line. The table below shows the average annual tuition and fees at four year colleges over a 12 year period. Year Cost (in Dollars) 1990 2035 1994 2820 1996 3151 1998 3486 2000 3774 2002 4273 Linear Models We first define x as the number of years since 1990. So in 1990, x=0, in 1994, x=4, etc. Then we can find the slope using the 2 end points of the data, (0, 2035) and (12,4273). Use the slope formula Year x Cost in dollars = y 1990 0 2035 1994 4 2820 1996 6 3151 1998 8 3486 2000 10 3774 2002 12 4273 Use slope intercept form Oct 7 10:58 AM Oct 7 11:22 AM 1 Now that we have found an equation to model the data, y=186.5x+2035, we can use the equation to predict what the cost of tuition and fees will be in 2016. Since x represents the number of years since 1990, substitute x=16 into the linear equation. We predict the cost of tuition and fees to be $5019 in 2014. Write a linear equation to model the data in the table. Year Travel Expenses 2001 500 2002 750 2003 1000 2004 1250 Oct 7 11:25 AM Oct 7 11:48 AM 9