Question

1. (ma.912.ar.4.3)

part a: use the table below to graph the absolute
value function.

x	-4	-2	0	1	3	4
y	0	4	8	6	2	0

part b: identify the domain and range of the
function. (write in inequality notation.)

1. (ma.912.f.1.5)

two linear functions, f(x) and g(x), are
represented below.
$f(x) = 2x - 2$
a. the y - intercept for both functions is the same.
b. the function $g(x)$ has a greater slope than $f(x)$.
c. both functions have the same domain and
range.
d. the rate of change for $f(x)$ is greater than the
rate of change for $g(x)$.

1. (ma.912.ar.4.3)

part a: graph the absolute value function below.
$f(x) = |x - 2| - 5$
part b: identify the vertex and all of the intercepts.
the vertex is (_, _).
the y - intercept is (_, _).
the x - intercepts are (_, _) and (_, _).

1. (ma.912.f.1.5)

two linear functions, $j(x)$ and $k(x)$, are represented
below.
$j(x) = -x + 5$
compare the two functions. select all that apply.

• the functions have the same y - intercept but

different slopes.

• the functions have different y - intercepts

but the same slope.

• the functions have the same x - intercepts.
• the x - intercept of $j(x) < k(x)$.
• the x - intercept of $j(x) > k(x)$.

Explanation:

Problem 3 Part B

Step1: Find the vertex of \( f(x) = |x - 2| - 5 \)

The general form of an absolute value function is \( f(x)=|x - h|+k \), where \((h,k)\) is the vertex. For \( f(x)=|x - 2|-5 \), \( h = 2 \) and \( k=-5 \), so the vertex is \((2,-5)\).

Step2: Find the y - intercept

To find the y - intercept, set \( x = 0 \) in the function \( f(x)=|x - 2|-5 \). Then \( f(0)=|0 - 2|-5=| - 2|-5 = 2 - 5=-3 \). So the y - intercept is \((0,-3)\).

Step3: Find the x - intercepts

To find the x - intercepts, set \( f(x)=0 \), so \( |x - 2|-5 = 0 \). Add 5 to both sides: \( |x - 2|=5 \). This gives two equations: \( x - 2 = 5 \) or \( x - 2=-5 \).

• For \( x - 2 = 5 \), add 2 to both sides: \( x=5 + 2=7 \).
• For \( x - 2=-5 \), add 2 to both sides: \( x=-5 + 2=-3 \).

So the x - intercepts are \((-3,0)\) and \((7,0)\).

Step1: Analyze the y - intercepts

• For \( f(x)=2x - 2 \), the y - intercept (when \( x = 0 \)) is \( f(0)=-2 \).
• From the graph of \( g(x) \), when \( x = 0 \), \( y = 2 \). So the y - intercepts are different. Option A is wrong.

Step2: Analyze the slopes

• The slope of \( f(x)=2x - 2 \) is \( m_f = 2 \).
• For \( g(x) \), using two points on the line (e.g., \((0,2)\) and \((1,4)\)), the slope \( m_g=\frac{4 - 2}{1 - 0}=2 \)? Wait, no, looking at the graph, when \( x = 0 \), \( y = 2 \), when \( x = 1 \), \( y = 4 \)? Wait, no, the graph of \( g(x) \): let's take two points. If we see, from the graph, when \( x=-2 \), the line starts, and when \( x = 0 \), \( y = 2 \), when \( x = 1 \), \( y = 4 \)? Wait, no, maybe I made a mistake. Wait, the function \( f(x)=2x - 2 \) has slope 2. Let's check the slope of \( g(x) \). Let's take two points: from the graph, when \( x = 0 \), \( y = 2 \); when \( x = 1 \), \( y = 4 \)? Then slope is \(\frac{4 - 2}{1 - 0}=2\)? But that can't be. Wait, no, maybe the graph of \( g(x) \): let's see, the line \( g(x) \) passes through \((0,2)\) and \((1,4)\), so slope is 2? But then option B: "The function \( g(x) \) has a greater slope than \( f(x) \)" is wrong. Wait, maybe I misread the graph. Wait, the graph of \( g(x) \): if we look at the grid, when \( x = 0 \), \( y = 2 \), when \( x = 1 \), \( y = 4 \), so slope is 2. But \( f(x) \) has slope 2. Wait, but maybe the graph is different. Wait, the problem says two linear functions \( f(x)=2x - 2 \) and \( g(x) \) (graph). Let's re - evaluate:

Wait, maybe the graph of \( g(x) \): let's take two points. Suppose when \( x = 0 \), \( y = 2 \), and when \( x = 1 \), \( y = 4 \), so slope is 2. But \( f(x) \) has slope 2. Then option B is wrong. Option C: Both functions are linear, so their domain is all real numbers (\( (-\infty,\infty) \)) and range is all real numbers (\( (-\infty,\infty) \)), so domain and range are the same. Option D: The rate of change (slope) of \( f(x) \) is 2, and if \( g(x) \) has slope 2, then D is wrong. Wait, maybe I made a mistake in the slope of \( g(x) \). Wait, maybe the graph of \( g(x) \) has a steeper slope. Let's check again. If the graph of \( g(x) \) goes from \( x=-2 \) (where \( y=-6 \) maybe) to \( x = 0 \), \( y = 2 \). Then the slope \( m_g=\frac{2-(-6)}{0 - (-2)}=\frac{8}{2}=4 \). Ah, that's the mistake. So \( f(x) \) has slope 2, \( g(x) \) has slope 4. So:

• Option A: y - intercept of \( f(x) \) is - 2, y - intercept of \( g(x) \) is 2, so A is wrong.
• Option B: Slope of \( g(x) = 4 \), slope of \( f(x)=2 \), so \( g(x) \) has greater slope. B is correct.
• Option C: Both are linear functions, domain is \( (-\infty,\infty) \), range is \( (-\infty,\infty) \), so C is correct.
• Option D: Rate of change of \( f(x) \) is 2, rate of change of \( g(x) \) is 4, so D is wrong.

So the correct options are B and C.

Step1: Analyze y - intercepts

• For \( j(x)=-x + 5 \), y - intercept (when \( x = 0 \)) is \( j(0)=5 \).
• For \( k(x) \), from the graph, when \( x = 0 \), \( y = 5 \). So y - intercepts are the same.

Step2: Analyze slopes

• Slope of \( j(x) \) is \( m_j=-1 \).
• For \( k(x) \), using two points \((0,5)\) and \((4,0)\), slope \( m_k=\frac{0 - 5}{4 - 0}=-\frac{5}{4}=-1.25\)? Wait, no, wait \( j(x)=-x + 5 \) has slope - 1, \( k(x) \): let's take two points, \((0,5)\) and \((4,0)\), slope is \(\frac{0 - 5}{4 - 0}=-\frac{5}{4}=-1.25\)? Wait, no, the slope of \( j(x) \) is - 1, slope of \( k(x) \): from the graph, when \( x = 0 \), \( y = 5 \), when \( x = 4 \), \( y = 0 \), so slope is \(\frac{0 - 5}{4 - 0}=-\frac{5}{4}=-1.25\)? Wait, no, the equation of \( k(x) \): using point - slope form, \( y - 5=\frac{0 - 5}{4 - 0}(x - 0)\), so \( y=-\frac{5}{4}x + 5 \)? Wait, no, the graph of \( k(x) \) passes through \((0,5)\) and \((4,0)\), so slope is \(-\frac{5}{4}\)? But \( j(x) \) has slope - 1. Wait, no, the problem says "Two linear functions, \( j(x) \) and \( k(x) \), are represented below. \( j(x)=-x + 5 \)" and the graph of \( k(x) \) passes through \((0,5)\) and \((4,0)\). So:
• Y - intercept: \( j(x) \) has y - intercept 5, \( k(x) \) has y - intercept 5, so same y - intercept. Slope of \( j(x) \) is - 1, slope of \( k(x) \) is \(\frac{0 - 5}{4 - 0}=-\frac{5}{4}=-1.25\)? Wait, no, \(\frac{0 - 5}{4 - 0}=-\frac{5}{4}\), which is - 1.25, but - 1 is greater than - 1.25. Wait, but let's check x - intercepts:
• For \( j(x)=-x + 5 \), set \( y = 0 \), \( 0=-x + 5\), \( x = 5 \). So x - intercept is \( (5,0) \).
• For \( k(x) \), from the graph, x - intercept is \( (4,0) \).

Now analyze the options:

• Option 1: "The functions have the same y - intercept but different slopes." Y - intercept is 5 for both, slope of \( j(x) \) is - 1, slope of \( k(x) \) is \(-\frac{5}{4}\) (different), so this is correct.
• Option 2: "The functions have different y - intercepts but the same slope." Y - intercepts are same, so wrong.
• Option 3: "The functions have the same x - intercepts." \( j(x) \) has x - intercept 5, \( k(x) \) has x - intercept 4, so wrong.
• Option 4: "The x - intercept of \( j(x)<k(x) \)." x - intercept of \( j(x) \) is 5, x - intercept of \( k(x) \) is 4, 5>4, so wrong.
• Option 5: "The x - intercept of \( j(x)>k(x) \)." 5>4, so this is correct.

So the correct options are the first and the fifth.

Answer:

The vertex is \(\boldsymbol{(2,-5)}\).
The y - intercept is \(\boldsymbol{(0,-3)}\).
The x - intercepts are \(\boldsymbol{(-3,0)}\) and \(\boldsymbol{(7,0)}\).

Problem 2