Lesson 1 Quick It’s Quadratic Develop Understanding

Ready

In the following exercise, you will be given the opportunity to recall what happens when you add two linear functions.

Given: $f (x) = 2 x + 3$ , $g (x) = x + 2$ , $h (x) = x - 5, m (x) = - 3 x - 1$

1.

$p (x) = f (x) + g (x)$

a.

Write the equation of $p (x)$ .

b.

In what way is the slope of $p (x)$ related to the slope of $f (x)$ and the slope of $g (x)$ ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.)

c.

In what way is the $y$ -intercept of $p (x)$ related to the $y$ -intercepts of $f (x)$ and of $g (x)$ ?

d.

Is $p (x)$ the equation of a line? Justify your answer.

2.

$q (x) = g (x) + h (x)$

a.

Write the equation of $q (x)$ .

b.

In what way is the slope of $q (x)$ related to the slope of $g (x)$ and the slope of $h (x)$ ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.)

c.

In what way is the $y$ -intercept of $q (x)$ related to the $y$ -intercepts of $g (x)$ and of $h (x)$ ?

d.

Is $q (x)$ the equation of a line? Justify your answer.

3.

$r (x) = f (x) + m (x)$

a.

Write the equation of $r (x)$ .

b.

In what way is the slope of $r (x)$ related to the slope of $f (x)$ and the slope of $m (x)$ ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.)

c.

In what way is the $y$ -intercept of $r (x)$ related to the $y$ -intercepts of $f (x)$ and of $m (x)$ ?

d.

Is $r (x)$ the equation of a line? Justify your answer.

4.

Graph $p (x)$ using the graphs of $f (x)$ and $g (x)$ that are shown in the graph.

5.

Graph $q (x)$ using the graphs of $g (x)$ and $h (x)$ that are shown in the graph.

6.

Graph $r (x)$ using the graphs of $f (x)$ and $m (x)$ that are shown in the graph.

7.

The graph of $s (x)$ is given. It is the sum of two of the given equations. It is your job to use what you know to determine which two functions were added to form $s (x)$ .

$s (x) =$ $+$ .

Set

Label the coordinates of the vertex and state whether it’s a maximum or a minimum. Then draw a dotted line for the line of symmetry. Write the equation for the line of symmetry.

8.

9.

10.

11.

12.

13.

14.

What connection exists between the coordinates of the vertex and the equation of the axis of symmetry?

15.

Look back at problem 13. Try to find a way to find the exact value of the coordinates of the vertex. Test your method with each vertex in problems 10–13. Explain your conjecture.

16.

How many $x$ -intercepts can a parabola have?

17.

Sketch a parabola that has no $x$ -intercepts, then explain what has to happen for a parabola to have no $x$ -intercepts.

18.

The equation of a quadratic function and the vertex of the parabola are given.

$f (x) = x^{2} - 2 x - 8$ ; vertex: $(1, - 9)$

a.

Is the vertex a maximum or a minimum?

b.

What is the line of symmetry of $f (x)$ ?

c.

What is the $y$ -intercept of $f (x)$ ?

d.

What is the domain of $f (x)$ ?

e.

What is the range of $f (x)$ ?

f.

On what intervals is $f (x)$ increasing?

g.

On what intervals is $f (x)$ decreasing?

Go

Rewrite the following expressions in an equivalent form without the square root.

19.

$\sqrt{49 a^{2} b^{6}}$

20.

$\sqrt{{(x + 13)}^{2}}$

21.

$\sqrt{9 m^{2} {(2 p^{3} - q)}^{2}}$