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In the last lesson, you learned about the shapes of the graphs and other properties of equations such as $y=2x^2-8x$. Now, you will learn about slightly different equations such as $y=2x^2-8x-2$.
Using the sliders, change the values of $a$, $b$, and $c$, and notice how each affects the shape and position of the graph of $y=ax^2+bx+c$.
Slide the slider for each of the coefficients $a$, $b$, and $c$ and observe what happens. Using your observations, note in the table below which coefficients affect the locations of the vertex, axis of symmetry, and roots of $y=ax^2+bx+c$.
Use the sliders to change the values of $a$, $b$, and $c$, and answer the following questions.
As you see, the value of $c$ doesn’t affect the axis of symmetry. This means that if you know that the axis of symmetry of $y=x^2+4x$ is the line $x=-2$, you also know that the axis of symmetry of $y=x^2+4x+3$ is the line $x=-2$.
Set $a=1$, $b=-4$, and $c=2$ to look at the graph of $y=x^2-4x+2$. Using the formula $$x=-b/{2a}$$, you can calculate that the axis of symmetry of this parabola is the line $x=2$. Also, notice that the vertex of this parabola is the point $(2,-2)$. Now slide $c$ to 4.5. The axis of symmetry of the parabola is still the line $x=2$, but the vertex has moved. The location of the vertex isn’t obvious from the graph, but you can find it algebraically: 1) The vertex is on the axis of symmetry, so its $x$-coordinate is 2. 2) The vertex is a point on the parabola, so it satisfies the equation for the parabola. This means that if we plug the $x$-coordinate of the vertex into the equation, we will get the $y$-coordinate: $y=x^2-4x+4.5$ $x=2$ → $y=(2)^2-4(2)+4.5=4-8+4.5=0.5$ So the vertex of this parabola is the point $(2,0.5)$.
So far you know that both $y=ax^2+bx+c$ and $y=a(x-h)^2+k$ have graphs which are parabolas. Let’s see which form gives you more information about the parabola and, knowing where the vertex of a parabola is, which form allows you to write an equation for the parabola more easily.
Click to get back a graph of $y=ax^2+bx+c$ with sliders for $a$, $b$, and $c$. Use the sliders and graph to check your answer.
Use the sliders to change the values of $a$, $h$, and $k$ and find an equation in the form $y=a(x-h)^2+k$ for a parabola with vertex at $(4,2)$.
Use the sliders to change the values of $a$, $b$, and $c$ to find an equation for a parabola in the form $y=ax^2+bx+c$ with vertex at $(4,2)$.
A quadratic equation in the form $\cl"red"{y=ax^2+bx+c}$ is said to be in standard form, while an equation in the form $\cl"blue"{y=a(x-h)^2+k}$ is said to be in vertex form. In this section you will learn to convert equations such as $y=2x^2-2x+3$ from standard form to vertex form.
In both forms $a$ is the coefficient of $x^2$, so the $a$’s in both forms are the same. $h$ and $k$ are the $x$- and $y$-coordinates of the vertex, so if you find the vertex of $y=2x^2-2x+3$, you will know $h$ and $k$.
Click . You will see the graph of $y_1=2x^2-2x+3$ in red and sliders for $a$, $h$, and $k$. Change the values of $a$, $h$, and $k$ to the ones you found.
Write $y=2x^2-2x+3$ in $y=a(x-h)^2+k$ form.