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In this lesson you will 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.
Using the table above, find the vertex and axis of symmetry of each parabola.
Here is the graph of $y=ax^2$. Use the slider to change the value of $a$ and answer the following questions.
Here are the graphs of $y_2=2x^2$ and $y=-2x^2$.
Complete this table.
Here is the graph of $y=ax^2$. Use the slider to change the value of $a$ to answer the following questions.
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.)
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.
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 $h$).
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 $y=a(x-h)^2+k$.