Have you ever thought about the speed of a baseball? Take a look at this dilemma.
When passing the baseball field, Mr. Travis handed the students the following problem written on a piece of paper that looked like a baseball. Here is what it said.
When an object is thrown into the air with a starting velocity of
feet per second, its distance
in feet, above its starting point
seconds after it is thrown is about
. Use a table of values to show the distance of an object from its starting point that has an initial velocity of 80 feet per second. Then graph the velocity of the ball.
To figure out this problem, you will need to know about quadratic functions and their graphs. Pay close attention because you will need to work with this problem again at the end of the Concept.
is a relation that assigns exactly one value of the domain to each value of the range.
When we say
, we are referring to any function that can be written in the form
are constants and
. This we defined as the standard form.
equal zero? What happens if it does? If the
value is zero, you might notice that it would make the first term
disappear because anything times zero is zero. You would be left with simply
. Although this is still a function, it is no longer quadratic. This is a linear function. All quadratic functions are to the
Let's look at quadratic functions in more detail.
You know that the word
refers to input values and the word
refers to output values. Recall that a
is a relation that assigns exactly one value of the domain to each value of the range. That means that for every
value, there is only one
We can find
values by substituting
values in the function. We organize the information using a table of values or a t-table. In most cases, the input values could be any numbers. However, for our convenience, we will use negative numbers, zero, and positive numbers.
Complete a table of values for the function
To find the
values, we will substitute the
values in the equation.
The completed t – table should look like this.
Evaluating a quadratic function is always the same. You substitute the
– values into the equation and solve the for
The values of
have an effect on the graphs of quadratic equations. Now we are going to use this information when we look at a quadratic function. What we know about the values of
help us to understand the opening of a parabola.
What it tells you
, graph opens upward
graph opens downward
is close to zero, wider graph
is far from zero, narrower graph
is less than zero so graph opens downward
is further from zero so will be narrow
helps predict axis of symmetry
axis of symmetry of the parabola
-axis at -2
We know that the graph of a quadratic function will always be a parabola.
is a kind of “U” shape that is always symmetrical on both sides.
It can go either upward or downward. Also, a parabola is not linear—no part of the parabola is actually a straight line. Thus, it cannot be vertical, either.
If we wanted to predict the shape of the parabola, we would need to look at the value of
. We know that
helps us to determine a parabola’s shape.
Take a look.
Now that you understand how these graphs look and how the equation of the graph affects its appearance, it is time to make some predictions.
What would you predict about the graph of
value is 7, it would be very narrow. Also, because
, it would open upward.
Answer the following questions by making predictions.
Predict the opening of
Solution: It will open downwards because the
value is negative.
For the quadratic function in Example A, where will the vertex be?
Which graph will have a wider opening one with a vertex at 0 or one with a vertex at 8?
Solution: Vertex at 0
Now let's go back to the dilemma from the beginning of the Concept.
First, think about the information that you have and the equation that you can write.
Next, we can make a table of values.
Finally we can take those values, insert them into a graphing calculator and create the following graph.
input value, independent value
output value, dependent value
relation that assigns one value of the domain to each value of range
degree in standard form-a parabola is created by a quadratic function.
Here is one for you to try on your own.
What would you predict about the graph of
, it would be very wide. Also, because
, it would open downward.
Quadratic Functions 1
Use your tables to graph the following functions.