Question

(a) graph the relationship \\( y = 15x^2 \\). to do so, plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. then click on the graph-a-function button.

(b) the length of the race is 240 meters. that is, the function \\( y = 15x^2 \\) produces an accurate distance \\( y \\) traveled by the car for \\( 0 \leq y \leq 240 \\).
fill in the blanks below to write an inequality for all the values of \\( x \\) for which the model produces an accurate value of \\( y \\). then choose the best description of this inequality.
inequality: \\( \square \leq x \leq \square \\)
description:
○ the portion of the domain of \\( y = 15x^2 \\) for which the modeled distance is accurate
○ the portion of the range of \\( y = 15x^2 \\) for which the modeled distance is accurate

(c) find the value of \\( x \\) when \\( y = 60 \\) according to the model. then choose the best description of this value of \\( x \\).
value of \\( x \\) when \\( y = 60 \\): \\( \square \\)
description:
○ the distance in meters the racecar travels after 60 seconds
○ the time in seconds the racecar needs to travel 60 meters
○ the time in seconds at which the racecars distance traveled in meters stops increasing

Explanation:

Step1: Find vertex of parabola

The vertex of $y=15x^2$ is at $(0,0)$ since it has no linear or constant term.

Step2: Find left/right points

Choose $x=-1$: $y=15(-1)^2=15$, so $(-1,15)$. Choose $x=1$: $y=15(1)^2=15$, so $(1,15)$. Choose $x=-2$: $y=15(-2)^2=60$, so $(-2,60)$. Choose $x=2$: $y=15(2)^2=60$, so $(2,60)$.

Step3: Solve inequality for valid x

Set $0 \leq 15x^2 \leq 240$. Divide all parts by 15: $0 \leq x^2 \leq 16$. Take square roots: $-4 \leq x \leq 4$.

Step4: Interpret inequality meaning

This represents valid input (domain) values for accurate distance.

Step5: Solve for x when y=60

Set $15x^2=60$. Divide by 15: $x^2=4$. Take square roots: $x=\pm2$.

Step6: Interpret x when y=60

$x$ represents time in seconds for 60 meters traveled.

Answer:

Part (a)

Plot the vertex $(0,0)$, left points $(-1,15), (-2,60)$, and right points $(1,15), (2,60)$ then draw the parabola.

Part (b)

Inequality: $-4 \leq x \leq 4$
Description: The portion of the domain of $y=15x^2$ for which the modeled distance is accurate

Part (c)

Value of $x$ when $y=60$: $\pm2$
Description: The time in seconds the racer needs to travel 60 meters