29 Quadratic Functions
Learning Objectives
In this section, you will:
- Recognize characteristics of parabolas.
- Understand how the graph of a parabola is related to its quadratic function.
- Determine a quadratic function’s minimum or maximum value.
- Solve problems involving a quadratic function’s minimum or maximum value.
Curved antennas, such as the ones shown in (Figure), are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
Recognizing Characteristics of Parabolas
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in (Figure).
The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of[latex]\,x\,[/latex]at which[latex]\,y=0.[/latex]
Identifying the Characteristics of a Parabola
Determine the vertex, axis of symmetry, zeros, and[latex]\,y\text{-}[/latex]intercept of the parabola shown in (Figure).
[hidden-answer a=”fs-id1165137892270″]
The vertex is the turning point of the graph. We can see that the vertex is at[latex]\,\left(3,1\right).\,[/latex]Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is[latex]\,x=3.\,[/latex]This parabola does not cross the[latex]\,x\text{-}[/latex]axis, so it has no zeros. It crosses the[latex]\,y\text{-}[/latex]axis at[latex]\,\left(0,7\right)\,[/latex]so this is the y-intercept.
[/hidden-answer]
Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions
The general form of a quadratic function presents the function in the form
where[latex]\,a,b,\,[/latex]and[latex]\,c\,[/latex]are real numbers and[latex]\,a\ne 0.\,[/latex]If[latex]\,a>0,\,[/latex]the parabola opens upward. If[latex]\,a<0,\,[/latex]the parabola opens downward. We can use the general form of a parabola to find the