Graphing in Standard Form
The standard form of a quadratic function is \(y=ax^2+bx+c\). Explore the graph of the function in standard form to the parent function \(y = x^2\) and answer the following questions.
The standard form of a quadratic function is \(y=ax^2+bx+c\). Explore the graph of the function in standard 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 \(b\) value affect the graph when compared to the parent function? a. If \(b > 0\)? b. If \(b < 0\)? 3. How does \(c\) value affect the graph when compared to the parent function? 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. When given a quadratic function in standard form, you need to solve for the vertex.
The x-coordinate of the vertex is found using the formula \(x=-\frac{b}{2a}\). (This is also the equation for the axis of symmetry).
The y-coordinate of the vertex is found by evaluating the quadratic function at the x-coordinate of the vertex.
We can say the vertex of the quadratic function in standard form is \(\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)\).
To graph a quadratic function in standard form:
- Plot the vertex.
- Draw in the axis of symmetry.
- Plot the y-intercept (0, c).
- Reflect the y-intercept 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 \(f\left(x\right)=3x^2-6x+1\). 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 graph).
Find the vertex and axis of symmetry
\(a=3\), \(b=-6\), \(c=1\)
\(x=-\frac{\left(-6\right)}{2\cdot3}\)
\(x=-\frac{\left(-6\right)}{6}\)
\(x=-\left(-1\right)\)
\(x=1\)
\(f\left(1\right)=3\left(1\right)^2-6\left(1\right)+1\)
\(f\left(1\right)=3-6+1\)
\(f\left(1\right)=-2\)
The vertex is \(\left(1,-2\right)\) and the axis of symmetry is \(x = 1\).
Find the y-intercept
The y-intercept is the point \(\left(0,1\right)\).
Find another point on the graph
Choose an \(x\) value. Let \(x = 2\)
\(f\left(2\right)=3\left(2\right)^2-6\left(2\right)+1\)
\(f\left(2\right)=3\cdot4-12+1\)
\(f\left(2\right)=12-12+1\)
\(f\left(2\right)=1\)
The point \(\left(2,1\right)\) is on the graph.
Plot these three points. Reflect the points \(\left(0,1\right)\) and \(\left(2,1\right)\) over the axis of symmetry.
The vertex is \(\left(1,-2\right)\).
The axis of symmetry is \(x=1\).
The domain is \(\left\{x\mid x\in R\right\}\) which can 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 \(y=-2\) when \(x=1\).
Example 2: Graph the quadratic function \(f\left(x\right)=-\frac{1}{3}x^2+4x-3\). Identify the vertex, axis of symmetry, domain, range, and maximum or minimum value.