FIND MAXIMUM OR MINIMUM OF A ABSOLUTE VALUE FUNCTION WORKSHEET
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Identify the vertex.
- Determine if the graph opens up or down.
- Determine if the graph has a maximum or minimum and its value.
- Decide if the graph is narrower, wider, or the same width as the parent graph.
Problem 1 :
y = -|x + 1|
Problem 2 :
y = 7|x - 3| - 4
Problem 3 :
y = -2/3|x - 1|
Problem 4 :
y = 5/2|x + 9| - 1
Problem 5 :
y = 3/4|x + 3| - 6
Problem 6 :
y = -|x| + 5
Problem 7 :
You are running a ten-mile race. The function d(t) = (1/8) ∣t − 40∣ represents the distance (in miles) you are from a water stop after t minutes.
a. Graph the function. Find the domain and range in this context.
b. Interpret the intercepts and the vertex. When is the function decreasing? increasing? Explain what each represents in this context.
Problem 8 :
Write the vertex of the absolute value function f(x) = ∣ax − h∣ + k in terms of a, h, and k.
Problem 9 :
Describe the transformation from the graph of f to the graph of g.
Answer Key
1) Vertex (-1, 0), open down, maximum at x = -1, width is same.
2) Vertex (3, -4), open up, maximum at x = 3, width is narrower.
3) Vertex (1, 0), open down, maximum at x = 1, width is wider.
4) Vertex (-9, -1), open up, minimum at x = -9, width is narrower.
5) Vertex (-3, -6), open up, minimum at x = -3, width is wider.
6) Vertex (0, 5), open down, maximum at x = 0, width is the same.
7) a) domain is t ≥ 0
the range is 0 ≤ d(t) ≤ 5
b) Distance covered in between 40 seconds is lesser than distance covered in between 40 to 80 seconds.
8) Vertex is at (h/a, k)
9) Then the value of k is -3.
Find x and y intercepts for the absolute value function given below.
Problem 1 :
y = 3|x - 1| + 2
Problem 2 :
y = 2|x|
Problem 3 :
y = |x| + 5
Problem 4 :
y = -2|x + 1| - 3
Problem 5 :
y = -3/5|x + 3| + 10
Problem 6 :
y = 15|x|
Problem 7 :
y = 5/3|x + 2| - 1
Problem 8 :
y = -2|x + 1| - 3
Problem 9 :
You ride your bicycle around a circular trail one time. The function
f(x) = (−1/3) ∣x − 4.5∣ + 1.5
represents the shortest distance (in miles) along the trail between you and your starting point after x minutes.
(a) Graph the function. Find the domain and range in this context.
(b) Interpret the intercepts and the vertex.
Answer Key
1)
there is no x-intercept.
y -Intercept is (0, 5).
2)
x -Intercept is (0, 0)
y -Intercept is (0, 0).
3)
The given function will not intersect the x-axis.
y -Intercept is (0, 5).
4)
There is no x-intercept.
y -Intercept is (0, -5)
5)
x-intercepts are (41/3, 0) and (-59/3, 0).
y -Intercept is (0, 41/5)
6)
x -Intercept is (0, 0).
y -Intercept is (0, 0).
7)
x -Intercept are (-7/5, 0) and (-13/5, 0).
y -Intercept (0, 7/3).
8)
There is no x-intercept.
y -Intercept is (0, -5).
9) a) the domain is {x | 0 ≤ x ≤ 9} and the range is { f (x) | 0 ≤ f (x) ≤ 1.5}.
b) The vertex (4.5, 1.5)
Find the following and graph it.
(i) Vertex
(ii) Slope
(iii) y-intercept and x-intercept
(iv) domain and range
(v) Increasing and decreasing interval.
Problem 1 :
f(x) = -3│x - 4│ + 3
Problem 2 :
f(x) = -1/2│x - 2│ + 4
Problem 3 :
f(x) = │x - 3│ - 2
Problem 4 :
f(x) = 3│x│
Problem 5 :
y = 3/5 │x│- 6
Problem 6 :
y =│x + 4│
1)
2)
3)
4)
5)
6)
Graph the following absolute value function by finding the following.
(i) Vertex
(ii) Slope
(iii) Direction of opening
(iv) x and y intercepts
(v) Domain and range
(vi) Increasing and decreasing
Problem 1 :
y = 3|x - 3|
Problem 2 :
y = -|x| + 4
Problem 3 :
y = (4/3) |x + 2| - 5
Problem 4 :
y = -(3/2) |x - 3| + 2
Problem 5 :
Match each function with its graph. Explain your reasoning.
i. f(x) = │x + 2│ − 2
ii. g(x) = −│x − 2│ + 2
iii. f(x) = −│x − 2│ − 2
iv. m(x) = │x + 2│ + 2
Answer Key
1) Vertex is at (3, 0)
Slope (a) = 3
Direction of opening = open up
x-intercept is at (3, 0)
y-intercept is at (0, 9).
- All real values is domain.
- Range is 3 ≤ y ≤ ∞
- To the left of minimum, it is decreasing.
- To the right of minimum, it is increasing.
2) Vertex is at (0, 4).
x-intercepts are (4, 0) and (-4, 0).
y-intercept is at (0, 4).
Slope (a) = -1
The curve will open down.
- All real values is domain.
- Range is 4 ≤ y ≤ -∞
- To the left of maximum, it is increasing.
- To the right of maximum, it is decreasing.
3) Vertex is at (-2, -5).
x-intercept is at (7/4, 0) and (-19/2, 0).
y-intercept (0, -7/3).
Slope (a) = 4/3
The curve will open up.
- All real values is domain.
- Range is 4 ≤ y ≤ -∞
- To the left of minimum, it is decreasing.
- To the right of minimum, it is increasing.
4) Vertex is at (3, 2).
x-intercept is at (13/3, 0) and (5/3, 0).
y-intercept is (0, -5/2).
Slope (a) = -3/2
The curve will open down.
- All real values is domain.
- Range is -5 ≤ y ≤ ∞
- To the left of maximum, it is increasing.
- To the right of maximum, it is decreasing.
5) i) Graph C is correct
ii) Graph B is correct.
iii) Graph D is correct.
iv) Graph A is correct.
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