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

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

Complete this table of values for the equations $y_1=x^2$ and $y=3x^2$. Note that $3x^2$ means $3(x^2)$ not $(3x)^2$: in an expression without parentheses, you raise a base to a power before multiplying it by another number. For example, when $x=2$, $3x^2=3(2^2)=3(4)=12$.

$x$ $y_1$ $y$

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

Equation	vertex	axis of symmetry
$y_1=x^2$		$x=0$
$y=3x^2$

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

Here is the graph of $y=ax^2$. Use the slider to change the value of $a$ and answer the following questions.

As $a$ increases, does the graph get narrower or wider?

As $a$ increases, does the axis of symmetry of the graph move left, move right, or stay in the same place?

As $a$ increases, does the vertex of the graph move left, move right, move up, move down, or stay in the same place?

Put $y=0.5x^2$, $y=3x^2$, $y=0.25x^2$, $y=7x^2$, and $y=2x^2$, in order from widest to narrowest.

What happens when $a$ is negative?

Here are the graphs of $y_2=2x^2$ and $y=-2x^2$.

Is the graph of $y=-2x^2$ narrower than the graph of $y_2=2x^2$, wider than that graph, or the same width as that graph?

Do $y_2$ and $y$ point in the same direction, or different directions?

If you flip the graph of $y_2$ across one of the coordinate axes, you will get the graph of $y$. Which coordinate axis is it?

The -axis

What is the largest value of $y$?

Can $y$ ever be positive?

Complete this table.

Equation	vertex	axis of symmetry
$y_2=2x^2$
$y=-2x^2$

Here is 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 $y=0.25x^2$, $y=2x^2$, 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?

Is the graph of $y=-3x^2$ narrower than the graph of $y=2x^2$, wider than that graph, or the same width as that graph?

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

Here is 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	$a$	$h$	$k$	vertex	axis of symmetry

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

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

As $a$ increases from $0$ to $5$, does the graph of $y$ become narrower or wider?

As $a$ decreases from $0$ to $-5$, does the graph of $y$ become narrower or wider?

If $a$ is positive, does the graph of $y$ open up or down?

If $a$ is negative, does the graph of $y$ open up or down?

Roots of quadratic equations

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.

Equation	Number of times it meets the $x$-axis

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

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?

Does the equation ever have more than 2 roots?

Does changing $h$ affect the number of roots of $y=a(x-h)^2+k$?

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 each possible number of roots.

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

Condition	Number of roots