Graphing in Vertex Form
The vertex form of a quadratic function is \(f\left(x\right)=a\left(x-h\right)^2+k\). Explore the graph of the function in vertex form to the parent function \(y = x^2\) and answer the following questions.
The vertex form of a quadratic function is \(f\left(x\right)=a\left(x-h\right)^2+k\). Explore the graph of the function in vertex form to the parent function \(y = x^2\) and answer the following questions.
Quick Check
1. How does the \(a\) value affect the graph when compared to the parent function? a. If \(a > 1\)? b. If \(0 < a < 1\)? c. If \(a < 0\)? 2. How does \(h\) affect the graph when compared to the parent function? a. If \(h > 0\)? b. If \(h < 0\)? 3. How does \(k\) affect the graph when compared to the parent function? a. If \(k > 0\)? b. If \(k < 0\)? Quick Check Solutions |
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To accurately graph a quadratic function, you probably want to plot at least five points, one of which is the vertex.
To graph a quadratic function in vertex form:
- Plot the vertex, which is \(\left(h,k\right)\).
- Draw in the axis of symmetry.
- Evaluate the function at any \(x\) value and plot the point.
- Reflect the point over the axis of symmetry.
- Evaluate the function at a different \(x\) value and plot the point.
- Reflect the point over the axis of symmetry.
Keep in mind, if \(a > 0\) (positive) the parabola opens up and if \(a < 0\) (negative) the parabola opens down.
Let’s look at a couple of examples.
Example 1: Graph the quadratic function \(f\left(x\right)=3\left(x+6\right)^2-1\). Identify the vertex, axis of symmetry, domain, range, and maximum or minimum value. (Some of the work is shown below, but watch the video to see the full explanation and actual graph).
Find the vertex and axis of symmetry
The vertex is \(\left(-6,-1\right)\) and the axis of symmetry is \(x = -6\).
Pick an x-value and evaluate the function
An \(x\) value closer to the axis of symmetry results in smaller \(y\) values to plot.
Choose an \(x\) value. Let \(x = -5\).
\(y=3\left(-5+6\right)^2-1\)
\(y=3\left(1\right)^2-1\)
\(y=3-1\)
\(y=2\)
The point \(\left(-5,2\right)\) is on the graph.
Repeat for another x-value
Let \(x=-4\)
\(y=3\left(-4+6\right)^2-1\)
\(y=3\left(2\right)^2-1\)
\(y=3\left(4\right)-1\)
\(y=12-1\)
\(y=11\)
The point \(\left(-4,11\right)\) is on the graph.
Plot these three points. Reflect the points \(\left(-5,2\right)\) and \(\left(-4,11\right)\) over the axis of symmetry.
The vertex is \(\left(-6,-1\right)\).
The axis of symmetry is \(x=-6\).
The domain is \(\left\{x\mid x\in R\right\}\), which can also be represented as \(\left(-\infty,\infty\right)\).
The range is \(\left\{y\mid y\ge-1\right\}\), which can also be represented as \(\left[-1,\infty\right)\).
There is a minimum at \(y=-1\) when \(x=-6\).
Example 2: Graph the quadratic function \(f\left(x\right)=-\frac{1}{2}\left(x-4\right)^2+3\). Identify the vertex, axis of symmetry, domain, range, and maximum or minimum value.