Chapter 2.4, Problem 61AYU

In Problems 64-71, use a graphing utility.

Exploration Graph $y=x^2$. Then on the same screen graph $y={\left(x-2\right)}^2$, followed by $y={\left(x-4\right)}^2$, followed by $y={\left(x+2\right)}^2$. What pattern do you observe? Can you predict the graph of $y={\left(x+4\right)}^2$? Of $y={\left(x-5\right)}^2$?

To determine

To graph: Graph the equations $y = x^2, y = {\left(x - 2\right)}^2, y = {\left(x-4\right)}^2$ and $y = {\left(x + 2\right)}^2$. Then observe the pattern from the graph drawn. Predict the graph of $y = {\left(x + 4\right)}^2$ and $y = {\left(x - 5\right)}^2$.

Answer to Problem 61AYU

Explanation of Solution

To sketch:

Graph the following equation with the graphing utility:

1. $y = x^2$

2. $y = {\left(x - 2\right)}^2$

3. $y = {\left(x - 4\right)}^2$

4. $y = {\left(x + 2\right)}^2$

Interpretation:

The equation $y = x^2$ is a parabola open upwards with the vertex point $\left(0, 0\right)$. Adding the value 2 with $x$ gives the shifted parabola with the vertex point $\left(-2, 0\right)$. And subtracting the values 2 and 4 gives the shifted parabola with the vertex point $\left(2, 0\right)$ and $\left(4, 0\right)$.

Therefore, the equation $y = {\left(x + 4\right)}^2$ gives the shifted parabola with the vertex point $\left(-4, 0\right)$ and the equation $y = {\left(x - 5\right)}^2$ gives the shifted parabola with the vertex point $\left(5, 0\right)$.

That is, the graph must be like follows: