Parabolas

In mathematics, a quadratic function is defined as any function that can be written in the form $f\left(x\right)={ax}^2+bx+c=0$ where a, b, and c are all real numbers and $a\ne 0$ . This is called the standard form of a quadratic function and produces a U-shaped curve called a parabola when graphed.

Here is an example of a parabola graphed on the Cartesian plane, $f\left(x\right)=x^2$ . Note that the quadratic function is in standard form because $a=1$ and both b and $c=0$ .

In this article, we'll explore the properties of parabolas including the nomenclature involved and how the a, b, and c affect the shape of the graph. Let's get started!

The vertex of a parabola

The vertex of a parabola is the highest or lowest point on the parabola depending on whether it opens upward or downward. In the example above, the vertex of $f\left(x\right)=x^2$ is $\left(0,0\right)$ or the origin point.

When $a>0$ in a quadratic function written in standard form, the parabola opens upward, and the vertex is the lowest point on the parabola. A very large positive value of a creates a narrow parabola, while a positive value of a closer to zero produces a wider parabola. The graph below provides an illustration of each:

When $a<0$ in standard form ${ax}^2+bx+c=0$ , the parabola opens downward and the vertex is the highest point on the parabola. A large negative value of a makes the parabola more narrow, while smaller values lead to a wider parabola. Again, here is an example of each:

Many questions will ask you for the vertex of a parabola, so this information will feel like second nature after a while.

The y-intercept of a parabola

While 'a' tells us about the shape and curve of a parabola, the 'c' for a quadratic function in the ' ${ax}^2+bx+c=0$ ' format tells us its y-intercept. More specifically, the 'c' value is the y-intercept, but remember to include an 'x' value of zero when you write it as an ordered pair. For instance, the y-intercept of ' $5x^2+7x-3$ ' would be $\left(0,-3\right)$ . Here are a few additional examples plotted on a graph:

The axis of symmetry of a parabola

The axis of symmetry of a parabola is defined as the straight line cutting through the vertex of a parabola and dividing it into two symmetric parts. We can determine the equation for the axis of symmetry for any quadratic function in standard form when $a\ne 0$ using the following formula:

$x=-\frac{b}{2a}$

In the image below, the red line represents the axis of symmetry:

Let's try looking at a sample problem. Find the axis of symmetry and the coordinates of the vertex for the quadratic function $-3x^2-6x+4$ . Our equation is already in standard form, so $a=-3,b=-6$ , and $c=4$ . Subbing -3 for a and -6 for b in the formula above, we get:

$\frac{-(-6)}{\left(2\right(-3\left)\right)}$

$x=-1$ is the axis of symmetry.

The axis of symmetry always cuts through the vertex of the parabola, which means -1 is the x-coordinate of the vertex. We get the y-coordinate by substituting -1 for x in our quadratic equation:

$-3{(-1)}^2-6(-1)+4=y$

$-3\left(1\right)-6(-1)+4=y$

$-3+6+4=y$

$7=y$

Now, we simply put the two values into an ordered pair to find the vertex: $\left(-1,7\right)$ . We're finished! There might be a lot of steps involved, but the math is doable as long as we take our time and work carefully.

Can a parabola open horizontally?

If we create a new expression by swapping the x's and y's, then we get a new $f\left(y\right)$ in terms of x. Put another way, we'll get a horizontal parabola with any quadratic function in the form $f\left(y\right)={ay}^2+by+c=0$ . Here is an example:

For parabolas like this, 'c' tells us the x-intercept instead of the y-intercept. If 'a' is positive, the graph opens to the right. If 'a' is negative, the parabola opens to the left.

That said, we don't see parabolas like this very often. Most of your problems will focus on ' $f\left(x\right)$ ' instead.

Practice questions on parabola

a. Given the quadratic function $f\left(x\right)=-2x^2+5x-7$ , which way will the parabola open?

The 'a' value of a quadratic formula in ${ax}^2+bx+c$ form determines which way it opens. In this case, 'a' is -2. Since -2 is less than zero, the parabola will open downward (∨ shape) and the vertex will be the highest point.

b. What is the y-intercept of the following quadratic function: $f\left(x\right)=9x^2+17x-4$ ?

The "c" value of a quadratic function in the form ${ax}^2+bx+c$ reveals its y-intercept. In our function above, the "c" value is -4. Therefore, the coordinates of the y-intercept would be $\left(0,-4\right)$ .

c. Find the equation for the axis of symmetry and the vertex of the following quadratic function: $4x^2+8x-6$ .

The first step in solving this problem is figuring out the equation for the axis of symmetry, which we can do with the following formula:

$-\frac{b}{2a}$

Our a value is 4 and our b value is 8, so let's substitute them into the formula:

$\frac{-8}{2\times 4}$

$x=-1$

The axis of symmetry always cuts through the vertex, so we now know that -1 is the x-coordinate of the vertex. We can find the y-coordinate by plugging -1 into our quadratic equation and solving for y:

$y=4{(-1)}^2+8(-1)-6$

$y=4\left(1\right)+8(-1)-6$

$y=4-8-6$

$y=-10$

With that, we know the y-coordinate of the vertex is -10. To answer the original question, $x=-1$ is the equation for the parabola's axis of symmetry, and the coordinates of the vertex are $\left(-1,-10\right)$ .

d. Which way will the following parabola open? $f\left(y\right)=-5y^2+6y+3$

Since this is a function of y instead of a function of x, our two options are opening to the right or the left. The a value is -5, so this parabola will open to the left because a is less than zero.

Topics related to the Parabolas

Conic Sections and Standard Forms of Equations

Latus Rectum

Eccentricity

Flashcards covering the Parabolas

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Parabolas

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

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