Back in Unit 1, you were asked to approximate the horizontal component when the vertical component was set equal to zero (find the x-intercepts). You were also asked to approximate the vertical component when the horizontal component was equal to zero (find the y-intercept). The former told us when our projectile hit the ground. The latter told us the height of the projectile at time t=0.
In this unit, we will use an algebraic approach to determine these values when we know the rule for the quadratic function in vertex form. Refer back to the data and rule for the projectile in unit 1. The rule for this data in function notation is . In this example, the "x"-intercepts correspond to the independent variable t. We are looking for the value of t (time) when the value of h (height) is equal to zero. Here is a graphical representation of the horizontal intercept furthest to the right.
As the above graph illustrates, when we are looking for a horizontal intercept (often referred to as an x-intercept), we are interested in the value of the horizontal component (time in this case) when the vertical component (height) is equal to zero. In other words, we are answering the real world question when is the projectile at height zero? The Graphing Calculator program used to produce the above graph has given us a very good approximation for the right side value. We can find both of the horizontal intercepts algebraically by substituting zero for h(t) in our function and solving the equation . Solve this equation algebraically for t. Solution
Again the question must be raised; are both of the values valid given our real world situation? Explain why or why not.
Another real world question we could ask is what is the height of the projectile at time t=0 (what is the y-intercept or h(t) intercept in this problem)? We will refer to this value as the initial height of the projectile.
Find the initial height of the projectile. Solution
Assuming time is in seconds and height is in meters in our problem, there are a couple of other questions that might be of interest to us:
1. Can you determind the height of the projectile at 3 seconds? 8 seconds?
2. Can you determine at what time(s) the projectile is at a height of 90 meters?
In order to answer the second question, we will need to use the graphing calclator until we develop some more algebraic skills.
Very often in text books, the variables x and y will be used for the independent and dependent variables respectively. For this reason, the horizontal intercepts are often referred to as x-intercepts, and the vertical intercept is referred to as the y-intercept. The x-intercepts may also be referred to as zeros (because we substitute zero for the dependent variable to find these values), roots (we will discuss roots and factors in a subsequent unit), or solutions (because we are solving a quadratic equation in order to find these values).
Here are several quadratic functions in vertex form. Write the ordered pair of the vertex and find the x and y-intercepts of these functions (if possible).
Check your work using your graphing calculator. Use 2nd TRACE (The CALCULATE feature). To find the x-intercepts, calculate the ZEROs. To find the y-intercept, you can use calculate VALUE and input x=0. You might also practice finding the vertex using the calculate MINIMUM or MAXIMUM feature.
Do all quadratic functions have a y-intercept?
Do all quadratic functions have two x-intercepts? EXPLORATION
By now, you should be fairly comfortable with the vertex form of a quadratic function. In the next few units, we will learn about an alternate form for a quadratic function that is more widely used in mathematics texts. We will learn how to extract all of the same information from this different looking equation.
Move on to Unit 6 - Standard Form of a Quadratic Function