How Do You Find Special Points on a Parabola?

Let’s look at how to use formulas for a parabola to get certain important points on a parabola.

Problem For the parabola y = 2x2 + 3x – 2, locate the points below.

a. The y-intercept.

Solution At the y-intercept, the x value is zero. This means that we need to set x = 0 in the equation:

y = 2(02) + 3(0) – 2 = -2

Putting this together, the y-intercept is at (0, -2).

b. The vertex.

Solution The vertex is located using the formula   where the values of a, b, and c come from the equation. In this case, a = 2, b = 3, and c = -2. This gives an x value on the intercept of

To find the corresponding y value, put this value into the equation,

This means the vertex is at (-3/4, –25/8).

c. The x-intercepts.

Solution At the x-intercepts, the y value is zero. Putting this into the equation yields

0 = 2x2 + 3x – 2

This equation is solved with the quadratic formula,

Put the values from the equation (a = 2, b = 3, and c = -2),

The x intercepts are at (-2, 0) and (1/2, 0).

All of these points are shown in the graph of the parabola below.

What are the Important Parts of a Parabola?

  • Green are the x intercepts that are solved with the quadratic formula. 
  • Purple is the y intercept found by setting x = 0.
  • Red is the vertex of the parabola. Since a > 0, the ends of the parabola point up and the vertex is a minimum. If a < 0, the vertex will be a maximum.

How Do You Graph a Formula in WolframAlpha?

A number of you have asked me about alternatives to a TI graphing calculator. In addition to Excel, there are many online graphing tools available. If you have an Android phone or IPhone, there are a huge number of free apps that are available. I have had a hard time finding just one that does everything we need. Most will graph formulas, but may or may not graph data and fit data to lines. None of them work identically to a graphing calculator which makes them difficult to support. But luckily there is another option! Continue reading “How Do You Graph a Formula in WolframAlpha?”

How Do You Interpret Inputs and Outputs for a Model?

In the example below, we want to look at the inputs and outputs for a function and interpret what they tells us. In both examples, the function is a quadratic function that models the rise and fall of an object thrown in the air.

Example 1 Suppose a ball is thrown into the air has its height (in feet) given by the function

\displaystyle h(t)=6+128t-16{{t}^{2}}

a. Find h(1) and explain what it means.

b. Find the height of the ball 4 seconds after it is thrown.

c. Test other values of  to decide if the ball eventually falls. When does the ball stop climbing?

 

Example 2 Suppose a ball is thrown into the air has its height (in feet) given by the function

\displaystyle h(t)=6+96t-16{{t}^{2}}

a. Find h(1) and explain what it means.

b. Find the height of the ball 3 seconds after it is thrown.

c. Test other values of  to decide if the ball eventually falls. When does the ball stop climbing?

The key thing is to test the function at enough points to convince yourself that the peak is really the peak. If the peak height occurs at x = 4.2, will finding h(3), h(4), and h(5) be enough to find that peak? If we have a graph, we can use it to find the peak. But if we only have the function we need to fine tune the input to zoom in on wherever the peak is.

In each case, the values of t are in seconds and h(t) is in feet. We want to find an h(t) value that is higher then those on either side.

How Are Inputs and Outputs Related Through A Model?

When working with a model, you need to pay careful attention to the units on each variable.

Problems 1 The number y (in millions) of women in the workforce is given by the function \displaystyle y=0.006{{x}^{2}}-0.018x+5.607 where x is the number of years after 1900.

a. Find the value of y when x = 44. Explain what this means.

b. Use the model to find the number of women in the workforce in 2010.

Solution for Problem 1.2.33 in text

The solution above is the correct strategy, but there is an error…can you find the error?