The graph of two functions, f(x) and g(x)=f(x+k)+k , is shown below. Determine the value of k
.
Two functions are graphed on a four quadrant coordinate plane. The horizontal x-axis goes from negative 6.5 to 9.5 in increments of 1 and the vertical y-axis goes from negative 9.5 to 6.5 in increments of 1. A V-shaped graph labeled f left parenthesis x right parenthesis that opens upward has a vertex at left parenthesis 0 comma 0 right parenthesis and passes through left parenthesis negative 4 comma 2 right parenthesis and left parenthesis 4 comma 2 right parenthesis. A second V-shaped graph labeled g left parenthesis x right parenthesis that opens upward has a vertex plotted and labeled at left parenthesis 3 comma negative 3 right parenthesis and passes through the plotted and labeled points left parenthesis negative 3 comma 0 right parenthesis, left parenthesis 0 comma negative StartFraction 3 over 2 EndFraction right parenthesis, and left parenthesis 9 comma 0 right parenthesis.
(1 point)
Responses
k=−3
k equals negative 3
k=−32
k equals negative 3 halves
k=3
k equals 3
k=9
To determine the value of k, we need to compare the graphs of f(x) and g(x) and find a relationship between them.
From the given information, we know that the vertex of g(x) is located at (3, -3), which means the vertex of f(x) is located at (3, -3-k).
Next, we can see that the point (-3, 0) lies on the graph of g(x), which means the corresponding point on f(x) is (-3, 0-k).
Using the information about the vertex and this corresponding point, we can determine the value of k.
If we substitute (-3, 0) into the equation of f(x), we get f(-3) = 0-k, which means f(-3) = -k.
Since the graph of f(x) passes through the point (-3, 0), f(-3) = 0. Therefore, we have 0 = -k.
Solving for k, we find k = 0.
So, the value of k is 0.
To determine the value of k, we need to compare the graphs of f(x) and g(x).
First, let's analyze the graph of f(x). It is a V-shaped graph that opens upward, with a vertex at (0,0). It passes through the points (-4,2) and (4,2).
Next, let's analyze the graph of g(x). It is also a V-shaped graph that opens upward, with a vertex at (3,-3). It passes through the points (-3,0), (0,-3/2), and (9,0).
The graph of g(x) is obtained by shifting the graph of f(x) horizontally by k units to the left and vertically by k units downward. Therefore, we need to find the value of k that would result in these shifts.
Comparing the vertices of the two graphs, we can see that f(x) has a vertex at (0,0) and g(x) has a vertex at (3,-3). The horizontal shift between the two vertices is 3 units to the left.
Comparing the points (-4,2) and (-3,0) on the two graphs, we can see that there is a horizontal shift of 1 unit to the right. This means that f(x) is shifted 1 unit to the right to obtain g(x).
Therefore, the total horizontal shift between the two graphs is 3 units to the left (from the vertex comparison) minus 1 unit to the right (from the comparison of points). This gives us a net horizontal shift of 2 units to the left.
Since k represents the horizontal shift of f(x) to obtain g(x), we can conclude that k = -2.
So, the value of k is -2.
The graph of g(x) is obtained by shifting the graph of f(x) to the left by k units and then shifting it up by k units.
Looking at the vertex of f(x) at (0, 0), we can see that the vertex of g(x) is at (3, -3). This means that the graph of g(x) is shifted to the left by 3 units and down by 3 units compared to the graph of f(x).
We also know that f(x) passes through the points (-4, 2) and (4, 2), and g(x) passes through the points (-3, 0) and (9, 0).
Since g(x) is a shifted version of f(x), we can determine the value of k by comparing the x-coordinates of the corresponding points on the two graphs.
The x-coordinate of (-3, 0) on g(x) corresponds to the x-coordinate of (-4, 2) on f(x), which means that g(-3) = f(-4). Since g(x) = f(x + k) + k, this can be written as:
f(-4) = f((-3) + k) + k
Similarly, the x-coordinate of (9, 0) on g(x) corresponds to the x-coordinate of (4, 2) on f(x), which means that g(9) = f(4). This can be written as:
f(4) = f((9) + k) + k
We can solve these equations to find the value of k.
f(-4) = f((-3) + k) + k
f(-4) = f(-3 + k) + k
f(-4) = f(-3 + k) + k
f(-4) = f(-3) + k
Since f(-4) = 2 and f(-3) = 0, we have:
2 = 0 + k
k = 2
f(4) = f((9) + k) + k
f(4) = f(9 + k) + k
f(4) = f(9) + k
Since f(4) = 2 and f(9) = 0, we have:
2 = 0 + k
k = 2
Therefore, the value of k is 2.