Modeling with Quadratic Functions

Standard Form of Quadratic Function

Recall a quadratic function in the standard form f(x) = ax2 + bx + c, where a is the leading coefficient, b is the middle term coefficient, and c is the constant (y-intercept) of the function. Those three values can tell a lot about the behavior of the function and the real-life scenario it models. You might also remember that a graph of a quadratic function is called a parabola. Interesting fact: parabola term comes from Greek, where it literally means ‘placing side by side’, just like the shape of the parabola is symmetric about the line in the middle, the axis of symmetry.

Graph of a Function in Standard Form

Let’s start with the leading coefficient a that determines how curved the parabola is. If the a value is positive, the parabola opens up, and the function has a minimum. Similarly, if a < 0, the parabola opens down and the function will have a maximum. The minimum or the maximum value of a quadratic function is also called the vertex (h,k), where h is the x-value of the vertex and k is the y-value of the vertex.

Vertex Form of Quadratic Function

Interestingly, we can rewrite a quadratic function using the vertex and the leading coefficient, and the resulting vertex form is f(x) = a(x – h)2 + k. Pay attention to the signs in the vertex form. For example, if the vertex is at (-3,2) and a = 1, the equation would be f(x) = (x + 3)2 + 2. However, if the vertex was at (3,-2), the signs in the equation would change and result in f(x) = (x – 3)2 – 2.

Graph of a Function in Vertex Form

Factored Form of Quadratic Function

The last form of a quadratic function that can be used to model a real-world scenario is factored form f(x) = a (x – r1)(x – r2), where r1 and r2 are the zeros (x-intercepts) of the function. Remember that the factors need to be set equal to zero and solve for x to ‘see’ the actual x-intercept.

For example, if f(x) = (x – 2)(x + 3), then x – 2 = 0 and x = 2. Similarly, x + 3 = 0, thus x = -3.

Graph of a Function in Factored Form

You can also graph any of the forms on a graphing calculator and read the key characteristics that are not visible from the equation!

Now that we have refreshed your memory about the quadratic forms and elements, let’s use them!

Application of Quadratic Function

Problem 1

You just bought yourself a brand new soccer ball and you want to know how high you can kick it. You know you kicked the ball from the ground level, which is the y-intercept and first zero at the same time (0,0). You estimated that the ball hit the ground approximately 60 feet from where you kicked it (that will be the second x-intercept). Your friend estimated that when the ball was 5 feet away from you, it was about 10 feet up in the air (additional point on the graph, (5,10)).

Now let’s figure out the given: zeros: (0,0) and (0,60), and another point (5,10).

Form to use: factored form y = a(x – r1)(x – r2)

Substitute the zeros for the r1 and r2 and the other point for x and y.

Unknown: the vertex and the a value

So we are going to have to solve for a in order to find the vertex.

As expected, the leading coefficient is negative, the ball will reach the maximum and hit the ground (obviously!) following a path of a parabola that opens down.

So now we can use the a value and the zeros to rewrite the factored form f(x) = -.036(x – 0)(x – 60) of the trajectory of the ball you kicked. We do not see the actual vertex in the factored form, but no fear, we have technology to help us out! Use any graphic calculator, handheld or online, to graph f(x) and find the highest y value either in the table or on a graph. Here is what you should see. The maximum height of the kick was approximately 32.4 feet.

Graph of the Kicked Ball

Problem 2