SAT Math: Quadratics & Parabolas

The focus of the new SAT Math test is on concepts of algebra more than it has been in the past. One of the things you’ll want to know well is how to work with quadratic equations and their graphs, which are parabolas. You’ll need to be able to solve quadratic equations by factoring and by using the Quadratic Formula. You’ll also need to know about the different forms of the equation for a parabola and about the properties of parabolas.

At the bottom of this post you’ll find a link to ten SAT practice problems involving quadratics and parabolas. There’s a second link down there to the solutions as well.

But first, let’s review some of the important characteristics of quadratic functions and their graphs.

Vertex Form
A quadratic function in vertex form will look like y = a(xh)² + k. In this form, the ordered pair (h, k) is the vertex of the parabola when the function is graphed, and the vertical line   x = h is the axis of symmetry. This axis of symmetry is a line that divides the graph in half down the middle, creating two pieces that are symmetrical (mirror images of each other).

Quadratics 5-17AConsider the equation y = (x – 3)² + 4. The vertex of the parabola will be the point with coordinates (3, 4) and the axis of symmetry will be the line x = 3.

 

 

 

 

 

Standard Form
When the equation of a quadratic function looks like y = ax² + bx + c, it is said to be in standard form. In this form, it’s not readily apparent where the vertex and the axis of symmetry are located. To find the axis of symmetry, you need a formula.

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

The vertex of the parabola must lie on the axis of symmetry, so once you have the equation of the axis of symmetry, you also have the x-coordinate of the vertex. To get the y-coordinate, just run that x-value through the equation.

Let’s look at an example.

Determine the equation of the axis of symmetry and the coordinates of the vertex for the parabola that results when the function y = 2x² – 8x + 11 is graphed in the xy-plane.

A.  x = -8; (-8, 203)
B.  x = -4; (-4, 75)
C.  x = 2; (2, 3)
D.  x = 4; (4, 11)

First, we need to find the axis of symmetry using the equation we saw above.

x=\frac{-b}{2a}=\frac{8}{2(2)}=\frac{8}{4}=2

So the axis of symmetry is the line x = 2. We also know that the x-coordinate of the vertex is 2 as well. To find the y-coordinate, simply run 2 through the equation.

y = 2(2)² – 8(2) + 11 = 3

So the vertex is (2, 3) and the correct answer is Choice C.

Direction
The value of a determines which way the parabola opens. If a is positive, the parabola opens upwards and if a is negative, the parabola opens downwards.

Quadratics 5-17B

 Symmetry
The symmetry property of parabolas means that each point on the parabola (other than the vertex) has a “mirror-image point” on the other side of the axis of symmetry. These points will be the same horizontal distance away from the axis of symmetry, and they will have the same y-coordinate.

Consider, for example, the graph of the function y = (x – 4)² + 1 graphed below.

Quadratics 5-17D

This function has a point at (1, 10) and another point at (7, 10). Each of these points are three horizontal units away from the axis of symmetry (x = 4) and they have the same y-coordinate. They are “mirror images” of each other.

OK, now that we’ve done a little review, it’s time for you to give those problems a try. Just click on the links below.

Additional Practice — Quadratics & Parabolas
Solutions — Quadratics & Parabolas

If you’d like to keep reviewing, we’ve published a book, “SAT Math: Focus on Quadratics & Parabolas,” that you can download to your Kindle, iPad, or tablet. It costs just $0.99 at Amazon.

If you have any questions about these problems or anything else on the SAT, send us an email at info@cardinalec.com .

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