Investigating $y=a(x-h)^2+k$

The objective of this lab is for you to learn about graphs of equations of the form $y=a(x-h)^2+k$. For example, you will look at equations such as $y=3x^2$, $y=-2x^2$, and $y=2(x+1)^2+3$ and compare them to $y=x^2$. You will also learn about roots of quadratic equations and how the values of $a$, $h$, and $k$ affect the number of roots.

Comparing $y=x^2$ to $y=3x^2$

Question 1. Complete this table of values for the equations $y_1=x^2$ and $y=3x^2$.
$x$ $y_1$ $y$
39
2
1
0
-1
-2
-327

Using the table above, find the vertex and axis of symmetry of each parabola.

Equationvertexaxis of
symmetry
$y_1=x^2$
$y=3x^2$

Click on to look at the graphs of $y_1=x^2$ and $y=3x^2$. Which parabola is narrower ($y_1$ or $y$)?

Question 2. Click on to see the graph of $y=ax^2$. Use the slider to change the value of $a$ and answer the following questions.

How do the shape, vertex and axis of symmetry of the graph change as $a$ increases in the equation $y=ax^2$?

Put $y=0.5x^2$, $y=3x^2$, $y=0.1x^2$, $y=7x^2$, and $y=2x^2$, in order from widest to narrowest.
$y=$$\,x^2$ (widest)
$y=$$\,x^2$
$y=$$\,x^2$
$y=$$\,x^2$
$y=$$\,x^2$ (narrowest)

What happens when $a$ is negative?

Question 3. Click on to see the graphs of $y_2=2x^2$ and $y=-2x^2$.

What is the relationship between $y_2$ and $y$?

What is the largest value of $y$?

Can $y$ ever be positive?

Complete this table.

Equationvertexaxis of
symmetry
$y_2=2x^2$
$y=-2x^2$

Which parabola, if any, is narrower?
$y_2=2x^2$
$y=-2x^2$
neither

Question 4. Click on to see the graph of $y=ax^2$. Use the slider to change the value of $a$ to answer the following questions.

We say that the graphs of $y=0.1x^2$, $y=2x^2$, and $y=3x^2$ open up. In what direction do the graphs of $y=-2x^2$, $y=-0.5x^2$, and $y=-3x^2$ open?

Which, if any, is narrower, $y=-3x^2$ or $y=2x^2$?
$y=-3x^2$
$y=2x^2$
neither

The vertex and axis of symmetry of $y=a(x-h)^2+k$

Question 5. Click on to see the graph of $y=a(x-h)^2+k$. Use the sliders to change the values of $a$, $h$, and $k$ so that you are looking at the graph of $y=2(x-1)^2+2$. Fill in the following table. (The equation for the graph is written below the grid.)

Equation$a$$h$$k$vertexaxis of
symmetry
$y=2(x-1)^2+2$ 2 1 2
$y=2(x+1)^2+3$ 2 −1 3
$y=-3(x+1)^2+3$

Question 6. Where is the vertex of the graph of $y=a(x-h)^2+k$?

What is the axis of symmetry of the graph of $y=a(x-h)^2+k$?

What is the effect of $a$ on the graph of $y=a(x-h)^2+k$? What makes it narrow, wide, open up, and open down?

Roots of quadratic equations

Question 7. Change the values of $a$, $h$ and $k$ so that you are looking at the graph of $y=-3(x+1)^2+3$.

Fill in the following table with how many times the graph of each equation meets the $x$-axis.

EquationNumber of times it
meets the $x$-axis
$y=-3(x+1)^2+3$
$y=2(x+1)^2+3$
$y=2(x-1)^2+2$

The places where the graph of an equation meets the $x$-axis are called the $x$-intercepts of the equation. We call the $x$-coordinates of these points the roots of the equation.

Question 8. You have seen that $y=-3(x+1)^2+3$ has 2 roots and $y=2(x+1)^2+3$ has no roots. Use the sliders to set $a=2$ and $h=1$. Use the slider for $k$ to change its value (without changing $a$ or $h$).

For what values of $k$ does the equation have 2 roots?

1 root?

No roots?

More than 2 roots?

Question 9. Use the slider for $h$ to change its value. What is the effect of changing $h$ on the number of roots of $y=a(x-h)^2+k$?

Question 10. Use the sliders to change the values of $a$, $h$, and $k$ and find an equation in the form $y=a(x-h)^2+k$ that has

2 roots: $y=$$\,(\,$$\,)^2$ 

1 root: $y=$$\,(\,$$\,)^2$ 

no roots: $y=$$\,(\,$$\,)^2$ 

Question 11. Complete this table to summarize the effects of $a$ and $k$ on the number of roots of $y=a(x-h)^2+k$.

ConditionNumber
of roots
$a≠0$ and $k=0$
$a>0$ and $k>0$
$a>0$ and $k<0$
$a<0$ and $k<0$
$a<0$ and $k>0$

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