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Expert-verified Found in: Page 566 ### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801 # Christopher is repairing the roof on a shed. He accidentally dropped a box of nails from a height of 14 feet. This is represented by the equation $h=-16{t}^{2}+14,$ where $h$ the height in feet and $t$ is the time in seconds. Describe how the graph is related to .

The graph of $h=-16{t}^{2}+14$ is the graph of $h={t}^{2}$ reflected across the $x$-axis and vertically stretched by a factor 16 and vertically translated 14 units up.

See the step by step solution

## Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^{2}+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

## Step 2. Define the transformations of the graph of the function.

(1) The graph of $-f\left(x\right)$ reflect the graph of $f\left(x\right)$ across the $x$-axis.

(2) The graph of $f\left(x\right)+c$ shifts the graph of $f\left(x\right),\text{\hspace{0.17em}\hspace{0.17em}}c$ units up.

(3) The graph of $k\text{\hspace{0.17em}}f\left(x\right)$, will vertically stretch the graph of $f\left(x\right)$ by a factor $k$ if $k>1$ and will vertically compress the graph of $f\left(x\right)$ by a factor $k$ if $0.

## Step 3. Describe how the graph of h=−16t2+14 is related to the graph of h=t2.

Observe the equation $h=-16{t}^{2}+14$.

The coefficient of role="math" localid="1647757558375" ${t}^{2}$ is negative, so the graph is reflected across the $x$-axis.

The graph is multiplied by $k=16$, so the graph is vertically stretched by a factor 16.

The constant term $c$ is 14, so the graph is translated 14 units up.

Therefore, the graph of $h=-16{t}^{2}+14$ is the graph of $h={t}^{2}$ reflected across the $x$-axis and vertically stretched by a factor 16 and vertically translated 14 units up. ### Want to see more solutions like these? 