Q30. - Algebra 1

Question

A ball is dropped off a cliff that is 100 feet high. The function $h=-16t^2+100$ models the height $h$ of the ball after $t$ seconds. Compare the graph of this function to the graph of $h=t^2$.

Step-by-Step Solution

Answer

The graph $\mathit{h}\mathbf{=}\mathbf{-}\mathbf{16}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{100}$ is the graph $\mathit{h}\mathbf{=}{\mathit{t}}^{\mathbf{2}}$ reflected across the x-axis and vertically stretched by a factor of 16 and vertically translated 100 units up. 

Step 1. State the concept .

$f\left(x\right)$Vertical Stretch and Vertical Compression:

$y=af\left(x\right),a>1$. Sketch the graph $f\left(x\right)$ vertically by a factor of a.

$y=af\left(x\right),0<a<1$. Compress the graph + vertically by a factor of a.

The graph $f\left(x\right)=a{x}^2+c$ stretches or compresses the parent graph vertically.

The coefficient of $x^2$ the term is negative, so the graph is reflected across the $x$-axis.

It $c$ is positive, the graph is translated up and if $c$ is negative, the graph is translated down from the parent graph.

Step 2. Graph each function .

The graph of the functions $h=t^2$ and $h=-16t^2+100$ is given by:

 

The red curve represents - $h=t^2$ and the blue curve represents -$h=-16t^2+100$.

Step 3. State the interpretation of the graph.

Since $a=16$ the graph is vertically stretched by a factor of 16. 

$c=100$, the graph is translated 100 units up from the graph $h=t^2$.

So, the graph $h=-16t^2+100$ is the graph $h=t^2$ reflected across the $x$-axis and vertically stretched by a factor of 16 and vertically translated 100 units up.