Attributes of Absolute Value Functions

X-intercept: the point(s) where the graph crosses the x axis, the value of y must = 0. This is also called roots, zeros, and solutions. You can graph and see where it crosses or set the equation = 0 and solve.

Y-intercept: the point where the graph crosses the y-axis, the value of x must = 0. find by graphing or plugging in 0 for x and solving.

Symmetry: A function has symmetry if a reflection or rotation carries its graph onto itself. The axis of symmetry for an absolute value function is always x = the x value of the vertex.

Maximum: the greatest y value for the interval of the x values. Found by graphing and finding the highest y value might have to use substitution to find.

Minimum: the smallest y value for the interval of the x values. Found by graphing and finding the lowest y value might have to use substitution to find.

Graph the absolute value function then answer the questions.

f(x) = |x – 1| - 3

x

y

The absolute value graph has been shifted down 3 and right 1.

Domain: (-∞,∞)Range: [-3,∞) since the absolute value graph is going up the vertex is at the bottom and the range goes to infinity from there.

Axis of symmetry is x = 1 because it is always x = x value of the vertex.

V (1,-3)

Graph the absolute value function then answer the questions.

f(x) = |x – 1| - 3

x

y

The y intercept can be found 2 ways: from the graph you can see that it crosses the y axis at (0,-2)Algebraically if you plug in 0 for x the answer is the y intercept: |0-1|-3=|-1|-3= 1-3 = -2So (0,-2)

The x intercept(s): can be found by looking at the graph as well (4,0) & (-2,0)However it can also be found algebraically: if you set the equation = to 0 and solve the answers you get are the x intercepts: | x – 1|- 3 = 0Get the absolute value by itself | x – 1| = 3 solve x – 1 = 3 and x – 1 = -3

x = 4 and x = -2Therefor (4,0) and (-2,0)

V (1,-3)

Graph the absolute value function then answer the questions.

f(x) = |x – 1| - 3

x

y

Last but not least minimum and maximum. Find the minimum and maximum on the interval [-1,3]

V (1,-3)

I drew bars to show you where you are looking for the highest and lowest point. Between these bars the highest point occurs at -1 and at 3 and the y value at both is -1 so -1 is the maximum. If you could not tell from the graph you could plug in the x where the highest point occurs and find it. The minimum occurs at 1 which has a y value of -3 so the minimum is -3

Graph the absolute value function then answer the questions.

f(x) = 2|x| + 1

x

y

The absolute value graph has been shifted up 1 and stretched by 2.

Domain: (-∞,∞)Range: [1,∞) since the absolute value graph is going up the vertex is at the bottom and the range goes to infinity from there.

Axis of symmetry is x = 0 because it is always x = x value of the vertex.

V (0,1)

Graph the absolute value function then answer the questions.

f(x) = 2|x| + 1

x

y

The y intercept can be found 2 ways: from the graph you can see that it crosses the y axis at (0,1)Algebraically if you plug in 0 for x the answer is the y intercept:2|0| + 1=0+1= 1 (0,1)

The x intercept(s): can be found by looking at the graph as well and as you see on the graph it never crosses the x axis so the the answer is NO SOLUTION.If you worked this out algebraically: 2|x| +1 = 0

|x| = -1/2 can’t solve no solutions

V (0,1)

Graph the absolute value function then answer the questions.

f(x) = 2|x| + 1

x

y

Last but not least minimum and maximum. Find the minimum and maximum on the interval [-1,3] V (0,1)

I drew bars to show you where you are looking for the highest and lowest point. Between these bars the highest point would occur at 3 and the y value if I plug in 3 for x would be 7 so 7 is the maximum. If you could not tell from the graph you could plug in the x where the highest point occurs and find it. The minimum occurs at 0 which has a y value of 1 so the minimum is 1

Some things to think about:

If you multiply the absolute value by a negative it reflects vertically so this would effect them maximum and minimum and range.

Since it is a function and the x value can not repeat there can only be 1 y intercept, but there can be 0, 1 or 2 x intercepts.

Domain is never effected and is always all real numbers.

You try finding the domain, range, intercepts, axis of symmetry, maximum and minimum on the interval [-1,3]

Here is a hard one:f(x) = - ½ | x + 1| - 2 x

y