Standard form of absolute value function

Question

Solution:

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The standard form of an absolute value function is f(x) = a|x - h| + k, which of the following represents the vertex?
Vertical Shift
Horizontal Shift
Stretch and Compression
How do you write an absolute value equation in standard form?

For example

We have to graph f(x) = |x - 1| + 5

By comparing it with general form f(x) = a |x - h| + k

As a is 1, the graph opens upwards with slope 1

As h is 1, k is 5, so the vertex is shifted 1 to the right and 5 from the origin.

Therefore, the standard form of an absolute value function is f(x) = a|x - h| + k is (h, k).

The standard form of an absolute value function is f(x) = a|x - h| + k, which of the following represents the vertex?

Summary:

(h, k) represents the vertex for the standard form of an absolute value function is f(x) = a|x - h| + k.

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from 0 on the number line.

The absolute value parent function, written as f(x)=|x|, is defined as

f(x)={x if x>00 if x=0−x if x<0

To graph an absolute value function, choose several values of x and find some ordered pairs.

x	y=\|x\|
−2	2
−1	1
0	0
1	1
2	2

Plot the points on a coordinate plane and connect them.

Observe that the graph is V-shaped.

(1) The vertex of the graph is (0,0).

(2) The axis of symmetry (x=0 or y-axis) is the line that divides the graph into two congruent halves.

(3) The domain is the set of all real numbers.

(4) The range is the set of all real numbers greater than or equal to 0. That is, y ≥0.

(5) The x-intercept and the y-intercept are both 0.

Vertical Shift

To translate the absolute value function f(x )=|x| vertically, you can use the function

g(x)=f(x)+k.

When k>0, the graph of g(x) translated k units up.

When k<0, the graph of g(x) translated k units down.

Horizontal Shift

To translate the absolute value function f(x)=|x| horizontally, you can use the function

g(x)=f(x−h).

When h>0, the graph of f(x) is translated h units to the right to get g(x).

When h<0, the graph of f(x) is translated h units to the left to get g(x ).

Stretch and Compression

The stretching or compressing of the absolute value function y=|x| is defined by the function y=a|x| where a is a constant. The graph opens up if a>0 and opens down when a<0.

For absolute value equations multiplied by a constant (for example,y=a| x|),if 0<a<1, then the graph is compressed, and if a>1, it is stretched. Also, if a is negative, then the graph opens downward, instead of upwards as usual.

More generally, the form of the equation for an absolute value function is y=a|x−h|+k. Also:

The vertex of the graph is (h,k).
The domain of the graph is set of all real numbers and the range is y≥k when a>0.
The domain of the graph is set of all real numbers and the range is y≤k when a<0.
The axis of symmetry is x=h.
It opens up if a>0 and opens down if a<0.
The graph y=| x| can be translated h units horizontally and k units vertically to get the graph of y=a|x−h|+k.
The graph y=a|x| is wider than the graph of y=|x| if |a|<1 and narrower if |a|>1.

#1

Is there such an equation? I know such equations exist for parabolas and circles.

Hypothetical Absolute Value Equation:

\(\displaystyle y = \mid x - h \mid + k\)

For vertex:

h = x

k = y

So

\(\displaystyle y = \mid x - 3 \mid + 1\)

or

\(\displaystyle y = \mid x - (+3) \mid + 1\) (under the hood)

Would be the graph of the an absolute function with it's vertex at

\(\displaystyle (3,1)\)

Keeping in mind that the absolute value equation (with vertex at \(\displaystyle (0,0)\)) is:

\(\displaystyle y = \mid x \mid\)

with an upside down triangle shaped graph (which can be shifted by changing the vertex).

Last edited: Oct 22, 2012

#2

Hello, Jason76!

You are correct! .Nice thinking process!

\(\displaystyle y \,=\,|x|\) is a \(\displaystyle \vee\)-shaped graph with its vertex at \(\displaystyle (0.0).\)

The general form might be: .\(\displaystyle y \:=\:a|x-h| + k\)
. . where the vertex is \(\displaystyle (h,k)\)
. . and \(\displaystyle a\) modifies the "spread" of the \(\displaystyle \vee.\)

#3

h:shifts function left or right

k: shifts function up or down

a: determines width on parabola and absolute value functions.

h or k only affects the vertex coordinates if the function has a vertex. Otherwise, each value (that you know) on the function has to be moved left, right, up, or down based on h and k.

Some different types of standard functon equations:

Absolute Value:

\(\displaystyle y = \mid x \mid\)

\(\displaystyle y = a\mid x \mid\)

\(\displaystyle y = a\mid x - h \mid + k\)

Square Root:

\(\displaystyle y = \sqrt{x}\)

\(\displaystyle y = a\sqrt{x}\)

\(\displaystyle y - a\sqrt{x -h} + k\)

Common Slanted Line:

\(\displaystyle y = (x)\)

\(\displaystyle y = a(x)\)