COLLEGE ALGEBRA: AN INVESTIGATIVE APPROACH

by

S. Kastberg & T. Barron


GRAPHING QUADRATIC FUNCTIONS


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1. Fill in the following chart using the given functions:

 x

 -3

 -2

 -1

 0

 1

 2

 3

 

             

 

             

 

             

 

2a. What do you notice about the range values of the functions and ? Is there any relationship between the values in your chart? What is it?

2b. How about the range values of the functions and ? Is there a relationship between the values in your chart? What is it?

2c. How about the range values of the functions and ? Is there a relationship between the values in your chart? What is it?

3. So if you have a table of values for and , where C is a positive real number, what is the relationship between the range values of the functions?

4. So if you have a table of values for and , where C is a negative real number, what is the relationship between the range values of the functions?

5. Using the graphing calculator on your computer or some other graphing tool, graph the three functions given in your table.

a.

b. Is there any similarity between the graph of this function and the graph of ? If so what is it?

c. Is there any similarity between the graph of this function and the graph of ? If so what is it?

6a. So in general, given a function f, whose graph is known (We knew the graph of above.) we can find the graph of f(x) + C, when C is a positive real number, by .

6b. We can find the graph of f(x) - C, when C is a negative real number, by.

7. Consider the graphs of the following functions, and type the color of the graph in the box.

a.

b.

c.

d.

8. Consider the graphs of the following functions, and type the color of the graph in the box.

a.

b.

c.

d.

 

Use what you learned in problem 7 to develop a general rule for graphing functions of the form f(x+c) when the graph of f is known.

When c is a positive number the graph of f is shifted c units to obtain the graph of f(x+c).

When c is a negative number the graph of f is shifted c units to obtain the graph of f(x+c).

VERTEX

9. The functions graphed above are called quadratic functions. The graph is called a parabola. Every parabola has a vertex. The turning point of the parabola is called the vertex. Consider the graphs below and find the vertex of each.

The red graph has vertex .

The purple graph has vertex .

The green graph has vertex .

The blue graph has vertex .

10. Consider the graphs below and find the vertex of each.

The red graph has vertex .

The purple graph has vertex .

The green graph has vertex .

The blue graph has vertex .

11. Now consider the graph of the function . (Use a graphing tool to plot the graph.)

What is the vertex of this parabola?

Is there a relationship between the vertex of the parabola and the equation ? If so what is it?

12. Use the relationship you discovered to find the vertices of the following parabolas.

a. vertex

b. vertex

c. vertex

 

DESCRIBING THE GRAPH OF THE PARABOLA

13. Find the y-intercept of .

14. Find the domain of .

15. Find the range of .

16. Use a graphing tool to graph the function . Follow the graph from the left most point to the right most point. If you trace the graph with your pencil, beginning at the left most point, what direction is your hand going before you reach the vertex? Up or Down?

What direction is your hand going after you pass the vertex? Up or Down?

17. These two sections of the graph are called decreasing and increasing.

a. What happens to the x-values of the points on the graph as you trace the graph ?

b. What happens to the y-values of the points on the graph before you reach the vertex?

c. What are the x-values that correspond to the y-values of the points on the graph before you reach the vertex? You can write your answer as an inequality or in interval notation. (Your answer will be the interval over which the function is decreasing.)

d. What happens to the y-values of the points on the graph after you pass the vertex?

e. What are the x-values that correspond to the y-values of the points on the graph after you pass the vertex? You can write your answer as an inequality or in interval notation. (Your answer will be the interval over which the function is increasing.)

18. Use the graphing tool on your computer to graph .

a. Find the y-intercept of .

b. Find the domain of .

c. Find the range of .

d. Find the vertex of .

e. is decreasing over .

f. is decreasing over .


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