Exercise 31 Page 600

We are asked to write an equation of the line on the logo. To do so let's use the given bullet point questions.

How can you find the coordinates of the points of intersection?

We know that the parabola and the line intersect when x= 0 and when x= 2. The points of intersection are the points common to both the parabola and the line. Therefore, to find the y-coordinates of these two points we will use the function that models the parabola. y=3x^2-4x+2 First, let's determine the y-value for x= 0!

y=3x^2-4x+2

Substitute

x= 0

y=3( 0)^2-4( 0)+2

▼

Simplify

CalcPow

Calculate power

y=3(0)-4(0)+2

ZeroPropMult

Zero Property of Multiplication

y=0+0+2

AddTerms

Add terms

y=2

The first point of intersection is (0,2). Let's find the other one!

y=3x^2-4x+2

Substitute

x= 2

y=3( 2)^2-4( 2)+2

▼

Simplify

CalcPow

Calculate power

y=3(4)-4(2)+2

Multiply

Multiply

y=12-8+2

AddSubTerms

Add and subtract terms

y=6

The second point of intersection is (2,6).

Can you write an equation of the line given the points of intersection?

Recall that exactly one line passes through two points. Therefore, given two points of intersection, we are able to write an equation of the line. We will write the equation of the line in slope-intercept form. y=mx+b Here, m is the slope and (0,b) is the y-intercept. Let's use the Slope Formula to find the slope m of the line on the logo. m=y_2-y_1/x_2-x_1 We will use the points of intersection, ( 0,2) and ( 2,6), as the points (x_1,y_1) and (x_2,y_2).

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 0,2) & ( 2,6)

m=6- 2/2- 0

SubTerms

Subtract terms

m=4/2

CalcQuot

Calculate quotient

m=2

The slope is equal to 2. Note that one of the points has an x-coordinate equal to 0, so it is the y-intercept of the line, (0,2). Therefore, b=2. Finally, we can write the equation of the line. y=2x+2