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In this lesson, we’ll go back to studying graphs of quadratic equations. You will learn about the shapes of the graphs of equations such as $y=x^2+x$, and how they relate to other properties of those equations.
Remember that the roots or zeros of an equation are the $x$-coordinates of the points where its graph touches the $x$-axis.
Also remember that the axis of symmetry of a parabola is the line that you could fold the parabola along so that its two sides would match each other.
Use the sliders to change the values of $a$ and $b$ in the equation $y=ax^2+bx$ and answer the following questions.
Complete the following table.
You can also find the roots of an equation without graphing, as in the following example. To find the roots and axis of symmetry of $y=x^2-3x$ without graphing, use factoring (or the distributive law), as follows.
The roots are where $y=0$, so we have $x^2-3x=0$ which factors into: $x(x-3)=0$. So, $x=0$ OR $x-3=0$ → $x=3$
Roots: The roots of $y=x^2-3x$ are 0 and 3.
Axis of symmetry: The axis of symmetry of a parabola passes midway between its roots, so the axis of symmetry of this parabola is the line $x=1.5$.
Use the sliders and graph to the left to check your answers.
In the next few questions, we will find the roots of the general equation $y=ax^2+bx$ with $a≠0$ by factoring, and use that to get a formula for the axis of symmetry of any equation in that form.
Your results from quadExs have been copied into the table below. Complete this table.
As you can see:
The axis of symmetry of $y=ax^2+bx$ is the line $$x=-b/{2a}$$.
Below you can see the graph of $y=x^2-6x$. The axis of symmetry of this parabola is the line:
We want to find the vertex of this parabola. The vertex is on the axis of symmetry, so its $x$-coordinate is 3. The vertex is also a point on the parabola, so it satisfies the equation for the parabola. This means that if you plug the $x$-coordinate of the vertex into the equation, you will get the $y$-coordinate.
Plugging 3 for $x$ into $y=x^2-6x$ gives $y=(3)^2-6(3)$ → $y=9-18$ → $y=-9$ So the vertex is at the point $(3,-9)$.
You can check your answers using the sliders and graph to the left.