What the Vertex Form of a Quadratic can tell you about the graph

The vertex form of a quadratic function:

y = a · (x - h) ² + k,

also tells you whether the graph of the quadratic is smiling or frowning. To check, simply look at the value of a, as you would if the equation had been written in standard form. If the value of a is positive then the quadratic is smiling and if the value of a is negative then the quadratic will be frowning.

The vertex form of a quadratic equation can also tell you about the location of the highest point (on a frowning quadratic) or the lowest point (on a smiling quadratic – see Figure 1 on the next page). This point (the highest point on a frowning quadratic or the lowest point on a smiling quadratic) is called the vertex.

The x-coordinate of the vertex is the number h that appears inside the parentheses of the vertex form and the y-coordinate of the vertex is the number k that appears outside the parentheses in the vertex form.

Figure 1: (a) In this quadratic, a = -1 and the shape of the graph is a “frown.” The vertex in this case is the highest point on the graph. (b) In this quadratic a = 0.5 and the shape of the graph is a “smile.” The vertex in this case is the lowest point on the graph.

 

Example

Figure 2 shows the graph of a quadratic function. Find the formula for this quadratic function and express the formula in vertex form and in standard form.

Figure 2: Find the formula of this quadratic function.

Solution

We will find the formula in vertex form (this is relatively easy as the x- and y-coordinates of the vertex are given) and then convert the vertex form to standard form.

The vertex form of a quadratic function has the format:

y = a · (x - h) ² + k,

where the letter h represents the x-coordinate of the vertex and the letter k represents the y-coordinate of the vertex.

Figure 2 shows that the x-coordinate of the vertex is equal to 2 and that the y-coordinate of the vertex is equal to 2. This means that the vertex form of this quadratic will be:

y = a · (x - 2) ² + 2.

All that remains is to find the numerical value of the constant a. To do this, you can use the x- and y-coordinates of any other point (i.e. other than the vertex) that lies on the quadratic – for example the point (4, 0) shown in Figure 3. To work out the value of a we will plug x = 4 and y = 0 into the vertex form and then solve for a.

0 = a · (4 - 2) ² + 2.

0 = a · 4 + 2.

-2 = a · 4 .

So, the equation for the quadratic function shown in Figure 2 (expressed in vertex form) is:

To convert this equation from vertex form to standard form, you can expand by FOILing and then collect like terms.

(Expand the (x – 2) 2 by FOILing)

(Multiply through by minus one half)

(Combine the like terms)

So, the equation for the quadratic function shown in Figure 2 (expressed in standard form) is: