Question

3.
which equation best represents the function?
a. ( h(x) = -3|x + 1| + 4 )
b. ( h(x) = 3|x - 1| - 4 )
c. ( h(x) = -3|x - 1| + 4 )
d. ( h(x) = 3|x + 1| + 4 )

1. the graph of the parent function ( f(x) = |x| ) is shown.

which graph shown the transformation of ( f(x) ) as ( g(x) = f(x) + 2 )?
a.
b.
c.
d.

1. a function ( g(x) ) has its vertex at ( (1, -3) ).

what is the effect of the transformation ( g(x - 4) ), and what is the new vertex?
a. horizontal shift 4 units right; new vertex ( (5, -3) )
b. horizontal shift 4 units left; new vertex ( (-3, -3) )
c. vertical shift 4 units down; new vertex ( (1, -7) )
d. vertical shift 4 units up; new vertex ( (1, 1) )

Explanation:

Question 3

Step1: Recall absolute value function form

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex and \(a\) determines the direction (positive opens up, negative opens down) and vertical stretch/compression.

Step2: Identify vertex from graph

From the graph, the vertex is at \((-1, 4)\)? Wait, no, looking at the grid, the vertex seems to be at \((-1, 4)\)? Wait, no, let's check the x and y coordinates. Wait, the first graph (question 3) has vertex at \(x=-1\)? Wait, no, maybe I misread. Wait, the options: let's check the vertex of each option.

For option a: \( h(x) = -3|x + 1| + 4 \), vertex at \((-1, 4)\), \(a=-3\) (opens down). But the graph shown has a V-shape opening up? Wait, no, the first graph (question 3) has a V-shape opening up? Wait, the graph in question 3: the left side is going down from the vertex, right side going up? Wait, no, the first graph (question 3) has a vertex, and the lines go up from the vertex? Wait, no, the y-axis: the vertex is at y=4, x= -1? Wait, no, maybe the vertex is at ( -1, 4)? Wait, no, let's check the options again.

Wait, option d: \( h(x) = 3|x + 1| + 4 \), vertex at (-1, 4), \(a=3\) (opens up). Let's check the graph: the vertex is at x=-1, y=4, and the lines go up, so the slope is positive. So the correct equation should have vertex at (-1, 4), \(a>0\) (opens up). So option d: \( h(x) = 3|x + 1| + 4 \). Wait, but let's confirm the vertex. The general form is \( y = a|x - h| + k \), so \( x - h = x + 1 \) implies \( h = -1 \), so vertex at (-1, 4). The graph in question 3 has vertex at (-1, 4), and since the lines are going up, \(a\) is positive. So the correct option is d.

Step1: Recall transformation of functions

The function \( g(x) = f(x) + 2 \) is a vertical shift of the parent function \( f(x) = |x| \) up by 2 units. The parent function \( f(x) = |x| \) has vertex at (0, 0). After shifting up 2 units, the vertex moves to (0, 2). Now, let's check the options:

Option B: The graph in option B has vertex at (0, 2), which is a vertical shift up by 2. Let's check the other options:

Option A: vertex at (0, -2) (shift down 2). Option C: vertex at (2, 2) (horizontal shift right 2). Option D: vertex at (-2, 0) (horizontal shift left 2). So the correct graph is B, which has vertex at (0, 2) and the same shape as \( f(x) \) shifted up 2.

Step1: Recall horizontal shift rule

For a function \( g(x) \), the transformation \( g(x - 4) \) is a horizontal shift. The rule is: \( g(x - h) \) shifts the graph of \( g(x) \) horizontally by \( h \) units. If \( h > 0 \), it's a shift to the right; if \( h < 0 \), to the left.

Step2: Apply the rule to the vertex

The original vertex of \( g(x) \) is (1, -3). For \( g(x - 4) \), we replace \( x \) with \( x - 4 \) in the function, so the vertex (h, k) of \( g(x) \) becomes (h + 4, k) because \( x - 4 = 0 \) when \( x = 4 \), so the new x-coordinate is \( 1 + 4 = 5 \), y-coordinate remains -3. So it's a horizontal shift 4 units right, new vertex (5, -3).

Answer:

d. \( h(x) = 3|x + 1| + 4 \)

Question 4