Question

review & refresh
write a linear function that relates y to x.
1.
2.

x	0	1.5	3	4.5
y	5	4	3	2

the vertices of a figure are given. draw the figure and its image after a dilation with the given scale factor. identify the type of dilation.

1. $a(-3,1), b(-1,3), c(-1,1); k=3$ 4. $j(2,4), k(6,10), l(8,10), m(8,4); k=\frac{1}{4}$

concepts, skills, & problem solving
comparing functions graph each equation. decide whether each graph represents a linear or nonlinear function. (see exploration 1, p. 295.)

1. $h = 5 + 6t$ equation 1

$h = 5 + 6t^2$ equation 2

1. $y = -\frac{x}{3}$ equation 1

$y = -\frac{3}{x}$ equation 2

identifying functions from tables does the table represent a linear or nonlinear function? explain.
7.

x	0	1	2	3
y	4	8	12	16

8.

x	6	5	4	3
y	21	15	10	6

identifying functions from equations does the equation represent a linear or nonlinear function? explain.

1. $2x + 3y = 7$
2. $y + x = 4x + 5$
3. $y = \frac{8}{x^2}$

identifying functions from graphs does the graph represent a linear or nonlinear function? explain.
12.
13.

Explanation:

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Problem 1:

Step1: Find slope using two points

Use points (2,0) and (0,-2):
$m = \frac{0 - (-2)}{2 - 0} = \frac{2}{2} = 1$

Step2: Identify y-intercept

From graph, $b = -2$

Step3: Write linear function

$y = mx + b$
$y = 1x - 2$
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Problem 2:

Step1: Calculate slope from table

Use $(0,5)$ and $(1.5,4)$:
$m = \frac{4 - 5}{1.5 - 0} = \frac{-1}{1.5} = -\frac{2}{3}$

Step2: Identify y-intercept

When $x=0$, $y=5$, so $b=5$

Step3: Write linear function

$y = -\frac{2}{3}x + 5$
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Problem 3:

Step1: Dilate each vertex by $k=3$

$A'(-3 \times 3, 1 \times 3) = (-9, 3)$
$B'(-1 \times 3, 3 \times 3) = (-3, 9)$
$C'(-1 \times 3, 1 \times 3) = (-3, 3)$

Step2: Identify dilation type

$k=3>1$, so it is an enlargement.
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Problem 4:

Step1: Dilate each vertex by $k=\frac{1}{4}$

$J'(2 \times \frac{1}{4}, 4 \times \frac{1}{4}) = (0.5, 1)$
$K'(6 \times \frac{1}{4}, 10 \times \frac{1}{4}) = (1.5, 2.5)$
$L'(8 \times \frac{1}{4}, 10 \times \frac{1}{4}) = (2, 2.5)$
$M'(8 \times \frac{1}{4}, 4 \times \frac{1}{4}) = (2, 1)$

Step2: Identify dilation type

$0---

Problem 5:

Step1: Analyze Equation 1

$h=5+6t$ is in $y=mx+b$ form: linear function (graph is straight line)

Step2: Analyze Equation 2

$h=5+6t^2$ has squared variable: nonlinear function (graph is parabola)
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Problem 6:

Step1: Analyze Equation 1

$y=-\frac{x}{3}$ is in $y=mx+b$ form: linear function (graph is straight line)

Step2: Analyze Equation 2

$y=-\frac{3}{x}$ has variable in denominator: nonlinear function (graph is hyperbola)
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Problem 7:

Step1: Check rate of change

$\frac{8-4}{1-0}=4$, $\frac{12-8}{2-1}=4$, $\frac{16-12}{3-2}=4$

Step2: Classify function

Constant rate of change: linear function
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Problem 8:

Step1: Check rate of change

$\frac{15-21}{5-6}=6$, $\frac{10-15}{4-5}=5$, $\frac{6-10}{3-4}=4$

Step2: Classify function

Non-constant rate of change: nonlinear function
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Problem 9:

Step1: Rewrite in standard linear form

$2x+3y=7$ can be rearranged to $y=-\frac{2}{3}x+\frac{7}{3}$, which is $y=mx+b$ form.

Step2: Classify function

Linear function (no exponents on variables other than 1)
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Problem 10:

Step1: Simplify the equation

$y + x = 4x + 5$ → $y=3x+5$, which is $y=mx+b$ form.

Step2: Classify function

Linear function (no exponents on variables other than 1)
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Problem 11:

Step1: Analyze variable exponent

$y=\frac{8}{x^2}=8x^{-2}$ has variable with exponent -2.

Step2: Classify function

Nonlinear function (variable has non-1 exponent)
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Problem 12:

Step1: Analyze graph shape

Graph is a straight line with constant slope.

Step2: Classify function

Linear function
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Problem 13:

Step1: Analyze graph shape

Graph is a curved line (hyperbola) with changing slope.

Step2: Classify function

Nonlinear function

Answer:

1. $y = x - 2$
2. $y = -\frac{2}{3}x + 5$
3. Dilated vertices: $A'(-9, 3)$, $B'(-3, 9)$, $C'(-3, 3)$; Type: Enlargement
4. Dilated vertices: $J'(0.5, 1)$, $K'(1.5, 2.5)$, $L'(2, 2.5)$, $M'(2, 1)$; Type: Reduction
5. Equation 1: Linear; Equation 2: Nonlinear
6. Equation 1: Linear; Equation 2: Nonlinear
7. Linear function (constant rate of change of 4)
8. Nonlinear function (changing rate of change)
9. Linear function (can be written in $y=mx+b$ form)
10. Linear function (simplifies to $y=3x+5$)
11. Nonlinear function (variable has exponent -2)
12. Linear function (graph is a straight line)
13. Nonlinear function (graph is a curved line)