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- Precalculus
- Polynomial and rational functions
- Quadratic functions
Key equations
general form of a quadratic function |
|
the quadratic formula |
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standard form of a quadratic function |
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Key concepts
- A polynomial function of degree two is called a quadratic function.
- The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
- The axis of symmetry is the vertical line passing through the vertex. The zeros, or
intercepts, are the points at which the parabola crosses the
axis. The
intercept is the point at which the parabola crosses the
axis. See
[link] ,
[link] , and
[link] .
- Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See
[link] .
- The vertex can be found from an equation representing a quadratic function. See
[link]
.
- The domain of a quadratic function is all real numbers. The range varies with the function. See
[link] .
- A quadratic function’s minimum or maximum value is given by the
value of the vertex.
- The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See
[link] and
[link] .
- Some quadratic equations must be solved by using the quadratic formula. See
[link] .
- The vertex and the intercepts can be identified and interpreted to solve real-world problems. See
[link] .
Section exercises
Verbal
What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?
If possible, we can use factoring. Otherwise, we can use the quadratic formula.
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Algebraic
For the following exercises, rewrite the quadratic functions in standard form and give the vertex.
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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