Question

At the state fair, Erin and her cousin ride the Ultra Drop roller coaster. When the ride plummets down the first hill, it dips below the loading platform. At the bottom, a camera snaps the riders picture before hurtling them back toward the sky. The equation y=x^2 - 9x + 20 models the roller coaster’s path over time. The variable y represents height (in feet) above or below the platform. At y=0, the roller coaster is even with the platform. The variable x represents the amount of time (in seconds) since the ride began. Factor and graph the equation to better understand Erin’s ride. Part 1: write the equation in factored form. Part 2: find the vertex of the parabola. Hint: to find the x-value of the vertex, take the average of the x-values of the x-intercepts or use the first part of the quadratic formula (x= - b/2a). Part 3: identify the intercepts.A. What are the x-intercepts? Hint: use the equation from Part 1.B. What is the y-intercept? Hint: use the equation y= x^2 - 9x + 20.Part 4: sketch the graph of y= x^2 - 9x + 20. Identify the vertex and x- and y- intercepts on your sketch. Part 5: use the graph to answer the questions.A. Between what times does the roller coaster dip below the platform?B. What is the height and time at which Erin’s picture is taken during the roller coaster ride? Hint: Erin’s picture is taken at the lowest point of the roller coaster.

Answer 1

1) y = (x -4)(x -5)

2) vertex (4.5, -0.25)

x value = 4.5

3) x intercept: 4 and 5

y intercept: 20

Explanation:

Part 1:

$\begin{gathered} y=x^2\text{ - 9x + 20} \\ y\text{ = }x^2\text{ - 5x -4x+ 20} \\ y\text{ =x (x}-5)\text{ -4(x -5)} \\ y\text{ = (x - 4)(x-5)} \end{gathered}$

Graphing the equation:

The equation in factored form:

$\begin{gathered} y=a(x-h)^2\text{ + k} \\ \text{where vertex (h, k)} \\ \text{vertex = (}4.5,\text{ -0.25)} \end{gathered}$

Part 2:

The vertex of the parabola is (4.5, -0.25)

The x value of the vertex:

$\begin{gathered} \text{The x intercepts are (}4,\text{ 0) and (5, 0)} \\ \text{The x values are 4 and 5} \\ \text{The average = }\frac{4+5}{2} \\ \text{The average = 9/2 = 4.5} \end{gathered}$

Part 3:

The x intercept is the point the y value is zero

y = x² - 9x + 20

0 = x² - 9x + 20

0 = (x - 4)(x -5)

(x -4) = 0 or (x - 5) = 0

x = 4 or x = 5

The x intercepts are 4 and 5

In ordered form: (4, 0) and (5, 0)

we were asked to se the equation:

The y intercept is the point the x value is zero

y = (0)² -9(0) + 20

y = 0 - 0 + 20

y = 20

the y-intercept is y = 20

In orderd form: (0, 20)