Lab Activity 10: Graphing Piecewise Functions - Victoria College

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Lab Activity 10: Graphing Piecewise Functions

SHOW ALL WORK AND JUSTIFY ALL ANSWERS.

Supplies needed: Pencil and highlighter/colored pencil/crayon

1. Consider the graph of the function f below and answer the following questions.

College Algebra

(a) Without knowing the exact function, what can we say about the points on this graph? As x increases, what happens to y?

(b) What is the domain of f ? (c) What is the range of f ? (d) Suppose the x-intercept is (2, 0). Will f (1) be positive or negative? Why? (e) Will f (5) be positive or negative? Why? (f) Does this function have a maximum value? (g) Does this function have a minimum value?

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2. Consider the piecewise function f (x) =

x2 + 1 if x < 0 3 - x if x 0

Let's break this into two pieces: g(x) = x2 + 1 and h(x) = 3 - x. Given the graphs below,

y = x2 + 1 Find: g(0) =

y = 3-x abc Find: h(0) =

Will g(0) be included when we graph f ? (Look at the restriction x < 0)

Will h(0) be included when we graph f ? (Look at the restriction x 0)

Plot the point (0, 1) with a big, open circle.

Plot the point (0, 3) with a big, closed circle.

Highlight the part of the curve where x < 0.

Highlight the part of the curve where x 0.

We use an open circle for endpoints of graphs of piecewise functions when we have < or >. We use a closed circle for endpoints of graphs of piecewise functions when we have or .

Let's put the highlighted pieces together to get the graph of f (x) =

x2 + 1 if x < 0 3 - x if x 0

Don't forget the open and closed circles at the points on each piece where x = 0. Isn't this exciting?!?!?!?!

c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Lab 10 ? Page 2

Name:

College Algebra

-x2 - 2x if x < -1

3. Consider the piecewise function f (x) = -x

if -1 x < 2

x-2

if x 2

Let's break this into 3 pieces: g(x) = -x2 - 2x, h(x) = -x, and k(x) = x - 2.

-x2 - 2x if x < -1

and add some labels to help us discuss each piece: f (x) = -x

if -1 x < 2

x - 2 if x 2

(Top piece) (Middle piece) (Bottom piece)

TOP PIECE: Let's start with g(x) = -x2 - 2x (or y = -x2 - 2x). (a) We know the graph of this function is a parabola. Does it open up or down?

(b) Find the domain. (c) Factor g(x) = -x2 - 2x. (HINT: Factor out a common monomial from every term).

(d) Find the x-intercepts. Write as points. Recall: An x-intercept is a point where a curve intercepts (touches or crosses) the x-axis. An x-intercept has a y-coordinate of 0.

( , ) and ( , )

(e) Find the y-intercept. Write as a point. Recall: A y-intercept is a point where a curve intercepts (touches or crosses) the y-axis. A y-intercept has an x-coordinate of 0.

(, )

(f) Find: g(-1) = Will this point be an open or closed circle in our piecewise graph? (Look at the restriction x < -1)

(g) Using the xy-table, plot function g(x) = -x2 - 2x. xy -3

-2

-1

0

1

2

(h) Plot the point (-1, 1) with a big, open circle. Highlight the part of the curve where x < -1. c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Lab 10 ? Page 3

MIDDLE PIECE: Now, consider h(x) = -x (or y = -x). (a) We know the graph of this function is a line. How did we know?

(b) Find the domain.

(c) What is the slope of this line?

(d) Find the x-intercept. Write as a point. (, )

Recall: To find the x-intercepts, set y = 0 and solve.

(e) Find the y-intercept. Write as a point. (, )

Recall: To find the y-intercept, set x = 0 and solve.

(f) Find: h(-1) = Will this point be an open or closed circle in our piecewise graph? (Look at the restriction -1 x < 2)

(g) Find: h(2) = Will this point be an open or closed circle in our piecewise graph? (Look at the restriction -1 x < 2)

(h) Using the xy-table, plot function h(x) = -x. xy -2

-1

0

1

2

(i) Plot the point (-1, 1) with a big, closed circle. Plot the point (2, -2) with a big, open circle. Highlight the part of the curve where -1 x < 2.

c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Lab 10 ? Page 4

Name:

College Algebra

BOTTOM PIECE: Now, consider k(x) = x - 2 (or y = x - 2).

(a) We know what the square root function looks like.

(b) Find the domain.

Recall: To find domain of functions with roots, look for even roots with variables in the radicand. Solve radicand 0.

.

The radicand is the part of the function that is under the radical.

(c) Find the x-intercept. Write as a point. (, )

(d) Explain why this function has no y-intercept.

(e) Find: k(2) = Will this point be an open or closed circle in our piecewise graph? (Look at the restriction x 2)

(f) Since our function has a square root, we can choose values of x so that our radicand x - 2 will be a

perfect square. This will make plugging values into an xy-table much easier.

Simplify the following:

02 =

12 =

22 =

32 =

The result of each is a perfect square. Now we can set the radicand equal to each of these perfect

squares.

(g) Solve:

x-2 = 0

x-2 = 1

x-2 = 4

x-2 = 9

We will use the x-values we justfound in our xy-table. Fill in the xy-table and plot y = x - 2.

xy 2

3

NOTE: 2 is the smallest number in the domain

. of k(x), so we adjust our xy-table. .

6 11

(h) Plot the point (2, 0) with a big, closed circle. Highlight the part of the curve where x 2. c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Lab 10 ? Page 5

Now, put the three pieces of the curves you highlighted together:

Now that wasn't so bad, was it?

With our graph, it is easy to see how to evaluate piecewise functions.

-x2 - 2x if x < -1

(Top piece)

f (x) = -x

if -1 x < 2 (Middle piece)

x - 2 if x 2

(Bottom piece)

Looking at the piecewise function f and its graph, circle which piece to use from the labels in the func-

tion above (Top/Middle/Bottom). Then evaluate the function at the given value.

NOTE: This is a function so you will plug each input value into ONLY ONE piece.

(a) f (-5) =

Top/Middle/Bottom

(b) f (-1) =

Top/Middle/Bottom

(c) f (0) =

1

(d) f

=

2

(e) f (2) =

Top/Middle/Bottom Top/Middle/Bottom Top/Middle/Bottom

(f) f (50) =

Top/Middle/Bottom

STEPS TO GRAPH A PIECEWISE FUNCTION:

(a) Sketch each piece separately. (b) For each piece, find the function values at each number given in a restriction for x. Decide if that

point will be included in the piecewise graph. (c) Highlight the parts of each graph that satisfy the conditions in the piecewise function. (d) Put the highlighted pieces together in a new graph, which is the graph of the piecewise function. (e) Remember to put open circles for the points on each piece where x < # or x > # and closed circles

for the points on each piece where x # or x #.

c Melanie Yosko This page may not be scanned, copied, or redistributed without permission. Lab 10 ? Page 6

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