What do you call the line that divides the parabola into two equal parts and passes through the vertex and the focus?

Learning Outcomes

• Identify the vertex, axis of symmetry, [latex]y[/latex]-intercept, and minimum or maximum value of a parabola from it’s graph.
• Identify a quadratic function written in general and vertex form.
• Given a quadratic function in general form, find the vertex.
• Define the domain and range of a quadratic function by identifying the vertex as a maximum or minimum.

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.

The [latex]y[/latex]-intercept is the point at which the parabola crosses the [latex]y[/latex]-axis. The [latex]x[/latex]-intercepts are the points at which the parabola crosses the [latex]x[/latex]-axis. If they exist, the [latex]x[/latex]-intercepts represent the zeros, or roots, of the quadratic function, the values of [latex]x[/latex] at which [latex]y=0[/latex].

Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown below.

Equations of Quadratic Functions

The general form of a quadratic function presents the function in the form

[latex]f\left(x\right)=a{x}^{2}+bx+c[/latex]

where [latex]a[/latex], [latex]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 equation for the axis of symmetry.

The axis of symmetry is defined by [latex]x=-\dfrac{b}{2a}[/latex]. If we use the quadratic formula, [latex]x=\dfrac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex], to solve [latex]a{x}^{2}+bx+c=0[/latex] for the [latex]x[/latex]-intercepts, or zeros, we find the value of [latex]x[/latex] halfway between them is always [latex]x=-\dfrac{b}{2a}[/latex], the equation for the axis of symmetry.

The figure below shows the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[/latex]. In this form, [latex]a=1,\text{ }b=4[/latex], and [latex]c=3[/latex]. Because [latex]a>0[/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\dfrac{4}{2\left(1\right)}=-2[/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\left(-2,-1\right)[/latex]. The [latex]x[/latex]-intercepts, those points where the parabola crosses the [latex]x[/latex]-axis, occur at [latex]\left(-3,0\right)[/latex] and [latex]\left(-1,0\right)[/latex].

The standard form of a quadratic function presents the function in the form

[latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]

where [latex]\left(h,\text{ }k\right)[/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.

Given a quadratic function in general form, find the vertex of the parabola.

One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[/latex], and where it occurs, [latex]h[/latex]. If we are given the general form of a quadratic function:

[latex]f(x)=ax^2+bx+c[/latex]

We can define the vertex, [latex](h,k)[/latex], by doing the following:

• Identify [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex].
• Find [latex]h[/latex], the [latex]x[/latex]-coordinate of the vertex, by substituting [latex]a[/latex] and [latex]b[/latex] into [latex]h=-\dfrac{b}{2a}[/latex].
• Find [latex]k[/latex], the [latex]y[/latex]-coordinate of the vertex, by evaluating [latex]k=f\left(h\right)=f\left(-\dfrac{b}{2a}\right)[/latex]

Find the vertex of the quadratic function [latex]f\left(x\right)=2{x}^{2}-6x+7[/latex]. Rewrite the quadratic in standard form (vertex form).

Given the equation [latex]g\left(x\right)=13+{x}^{2}-6x[/latex], write the equation in general form and then in standard form.

Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all [latex]y[/latex]-values greater than or equal to the [latex]y[/latex]-coordinate of the vertex or less than or equal to the [latex]y[/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.

The domain of any quadratic function is all real numbers.

The range of a quadratic function written in general form [latex]f\left(x\right)=a{x}^{2}+bx+c[/latex] with a positive [latex]a[/latex] value is [latex]f\left(x\right)\ge f\left(-\frac{b}{2a}\right)[/latex], or [latex]\left[f\left(-\frac{b}{2a}\right),\infty \right)[/latex]; the range of a quadratic function written in general form with a negative [latex]a[/latex] value is [latex]f\left(x\right)\le f\left(-\frac{b}{2a}\right)[/latex], or [latex]\left(-\infty ,f\left(-\frac{b}{2a}\right)\right][/latex].

The range of a quadratic function written in standard form [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex] with a positive [latex]a[/latex] value is [latex]f\left(x\right)\ge k[/latex]; the range of a quadratic function written in standard form with a negative [latex]a[/latex] value is [latex]f\left(x\right)\le k[/latex].

How To: Given a quadratic function, find the domain and range.

1. The domain of any quadratic function as all real numbers.
2. Determine whether [latex]a[/latex] is positive or negative. If [latex]a[/latex] is positive, the parabola has a minimum. If [latex]a[/latex] is negative, the parabola has a maximum.
3. Determine the maximum or minimum value of the parabola, [latex]k[/latex].
4. If the parabola has a minimum, the range is given by [latex]f\left(x\right)\ge k[/latex], or [latex]\left[k,\infty \right)[/latex]. If the parabola has a maximum, the range is given by [latex]f\left(x\right)\le k[/latex], or [latex]\left(-\infty ,k\right][/latex].

Find the domain and range of [latex]f\left(x\right)=-5{x}^{2}+9x - 1[/latex].

Find the domain and range of [latex]f\left(x\right)=2{\left(x-\dfrac{4}{7}\right)}^{2}+\dfrac{8}{11}[/latex].

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The axis of symmetry of a parabola is the line that runs through the center of the parabola and is perpendicular to the focus. The focus is the point on the axis of symmetry that is closest to the vertex.

What is an Axis of Symmetry?

An axis of symmetry is an imaginary line that passes through the center of a two-dimensional figure. Every point on the line has equal distances from the two sides of the figure. In a parabola, the axis of symmetry is always perpendicular to the directrix and always goes through the focus.

Axis of Symmetry Definition

The axis of symmetry of a parabola is the line that runs through the vertex and is perpendicular to the line of symmetry. The axis of symmetry can be found by finding the equation of the line of symmetry and solving for x. The axis of symmetry is the x-coordinate of the vertex.

Axis of Symmetry Equation

An axis of symmetry is an imaginary line that passes through the center of a parabola and bisects it into two equal halves. The equation for the axis of symmetry of a parabola can be found by using the following formula:

axis of symmetry = x-coordinate of vertex

So, if the coordinates of the vertex of a parabola are (h, k), then the equation for its axis of symmetry would be:

axis of symmetry = h

Axis of Symmetry Formula

The axis of symmetry of a parabola is the line that divides the parabola into two symmetrical halves. The axis of symmetry is perpendicular to the directrix and passes through the vertex of the parabola. The equation of the axis of symmetry is y = -b/2a.

Standard form

A parabola is a two-dimensional curve that can be described by the equation y = ax^2 + bx + c. The axis of symmetry of a parabola is the line that divides the curve into two mirror images. This line is perpendicular to the directrix of the parabola, and it passes through the vertex of the parabola.

Vertex form

The vertex form of a parabola is given by the equation:

y = a(x – h)^2 + k

where (h, k) is the vertex of the parabola. The axis of symmetry of the parabola is the line x = h.

What is a Parabola?

A parabola is a two-dimensional, symmetrical curve, which is defined by a quadratic equation in standard form. The axis of symmetry of a parabola is the line that runs through the midpoint of the vertex and the focus. The focus is the point on the parabola where the light rays reflect off the surface.

Derivation of the Axis of Symmetry for Parabola

A parabola can be defined as a two-dimensional, symmetrical curve that is generated by the set of points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) not on the directrix.

The axis of symmetry of a parabola is the line that passes through the focus and is perpendicular to the directrix. The axis of symmetry divides the parabola into two mirror-image halves.

To derive the equation for the axis of symmetry, we will use the fact that the distance between any point on the parabola and the focus is equal to the distance between that point and the directrix.

Let P be any point on the parabola with coordinates (x,y). We can then write:

(x – x_0)^2 + (y – y_0)^2 = (x – p)^2 ………………………..(1)

where (x_0,y_0) are the coordinates of the focus and p is the parameter of the directrix.

How to Find the Axis of Symmetry of a Parabola

A parabola is a symmetrical curve, which means that it has an axis of symmetry. The axis of symmetry is the line that divides the parabola into two halves that are mirror images of each other.

To find the axis of symmetry of a parabola, we need to find its vertex. The vertex is the point on the parabola where the curve changes direction. It is also the point where the axis of symmetry intersects the parabola.

We can find the vertex by solving the quadratic equation that defines the parabola. Once we have found the vertex, we can use it to find the axis of symmetry. Theaxis of symmetry is perpendicular to the line that passes through the vertex and focus of the parabola.

Examples

A parabola is a two-dimensional curve that can be described by the equation y = x2. The line of symmetry of a parabola is the line that divides the curve into two mirror images. The axis of symmetry is the line that passes through the vertex of the parabola and is perpendicular to the line of symmetry.

The following are examples of parabolas and their axes of symmetry:

• y = x2 has an axis of symmetry at y = 0.

• y = -x2 has an axis of symmetry at y = 0.

• y = (x – 1)2 has an axis of symmetry at x = 1.

• y = (x + 2)2 has an axis of symmetry at x = -2.

Conclusion

As you can see, the axis of symmetry of a parabola is a very important concept. It allows us to determine the location of the vertex and the focus, as well as the direction of the parabola. If you understand how to find the axis of symmetry of a parabola, you’ll be well on your way to understanding this important curve.