Q. 20 - Precalculus Enhanced with Graphing Utilities

Question

Graph the function $f$ by starting with the graph of $y=x^2$ and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility.

$f\left(x\right)=2x^2+4$

Step-by-Step Solution

Answer

The required graph is shown below:

Step 1. Given information.

The given function is:

$f\left(x\right)=2x^2+4$

Step 2. Determine the transformations used.

In the function $f\left(x\right)=a(x-h{)}^2+k$, $a$ is a constant and $(h,k)$ is the vertex.

In the given function $a=2,h=0,k=4$. It means the graph of the given function is a parabola that opens up and has its vertex at (0,4) and its axis of symmetry is the line $x=0$.

The graph of $y=x^2$ is vertically stretch by factor 2 and shifts 4 units up.

First, plot the graph of $y=x^2$ then multiply the values by 2 to get the graph of $y=2x^2$, then add 4 in the values to get the graph of the function $f\left(x\right)=2x^2+4$.

Step 3. Draw the graph.

The required graph is shown below:

Using a graphing utility, we get a graph that is the same as the above graph.