Question

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 $h=-16t^2+14,$ where $h$ the height in feet and $t$ is the time in seconds. Describe how the graph is related to .

Short answer

The graph of $h=-16t^2+14$ 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.

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 $f\left(x\right),c$ units up.

(3) The graph of $kf\left(x\right)$, 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 $0<k<1$.

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

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

The coefficient ofrole="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$h=-16t^2+14$ 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.