Graph a Line using Intercepts Recall: Intercepts the points where a relation crosses the axes. Example 1 : A graph has an xintercept of 4 and a yintercept of 5. a) What are the coordinates of these two points? As a point, (4,0) x-intercept ← As a point, (0,5) y-intercept ← b) Use the slope formula to determine the slope of this line. 𝑚 = 𝑦 2 −𝑦 1 𝑥 2 −𝑥 1 = 5−00−4 = 5 −4 =− 54 c) What is the equation of this line? _____ 𝑦 = 𝑚𝑥 + 𝑏, 𝑚 =− 54 , 𝑏 = 5 𝑦 =− 5 4𝑥 + 5 Example 2 : A line has an xintercept of 6 but no yintercept. Describe this line and writ e an equation.
Sol: x-intercept of -6, as a point (-6,0). Since there is no 3-intercept (this means that the line never crosses the y-axis), then this is a vertical line. The equation of the line is x=-6. Example 3 : a) Determine the x and y-intercepts for the line: 3x-4y=12. x-intercept: Set y=0 and solve 3x-4y=12 for x. 3x-4(0)=12 3x-0=12 3x=12 3x/3=12/3 x=4 So, the x-intercept is 4. Or as a point (4,0) y-intercept: Set x=0 and solve 3x-4y=12 for y. 3(0) − 4𝑦 = 12 0 − 4𝑦 = 12 − 4𝑦 = 12 −4𝑦 −4 = 12 −4 y=-3 So, the y-intercept is -3 or as a point (0,-3) b) Graph this line using the intercepts. -Plot the x and y-intercepts
-Join them with a straight line Example 4 : The equation 4x+6y-48=0 represents the number of comic books and novels that can be purchased with a given amount of money. Comic books cost $4 each and novels cost $6 each. a) What does x represent? x represents the number of comic books. b) What does y represent? y represents the number of novels. c) What is the x-intercept? What does it represent? Set y=0 and solve 4x+6y-48=0 for x. 4𝑥 + 6(0) − 48 = 0 4𝑥 − 48 = 0 4𝑥 = 0 + 48 4𝑥 = 48 4𝑥 4 = 48 4 x=12 So, the x-intercept is x=12 or as a point (12,0) The x-intercept represents the number of comic books we can buy with $48. d) What is the yintercept? What does it represent?
Set x=0 and solve 4x+6y-48=0 for y. 4(0) + 6𝑦 − 48 = 0 6𝑦 − 48 = 0 6𝑦 6 = 48 6 𝑦 = 8 So, the y-intercept is 8 and it represents the number of novels we can buy with $48.00 e) What amount of money can this person spend? 4x+6y=48 This person can spend $48.00 f) Graph this relation using the intercepts. Why do we use only the first quadrant? - Plot the intercepts -Join them with a straight line We use the first quadrant only since we cannot buy a negative number of books g) The point (3/2, 7) is on this line. What does this point mean? Is this possible? - This means that we can buy 1.5 comic books and 7 novels with $48.00 - No, it is not possible to buy 1.5 comic books