4.4 – Absolute Value Equations

What is the absolute value of a number?

Example 1 – Simplify a) |6| b) |−4| c) |−7

3|

Example 2 – Solve |𝑥| = 2

Steps for solving an absolute value equation:

1) Get the absolute value by itself on one side (everything not in the absolute value should be on the other side).

2) Set up two cases: the positive case and the negative case. Solve for each case. 3) Check each solution to see if it is an actual or extraneous solution.

Example 3 – Solve |𝑥 + 7| = 10First, by inspection:

Example 4 – Solve |𝑥 − 2| + 3 = 9

Positive Case: Negative Case: Check:

Positive Case: Negative Case: Check:

Example 5 – Solve |3𝑥 − 2| = 1 − 𝑥 algebraically

Example 6 – Solve |𝑥 − 3| + 7 = 4

Example 7 – Solve |4𝑥 − 5| + 2 = 2

Check:

Check:

An Absolute Value Equation with No Solution: no solutions

Example 8 – Solve |𝑥 + 5| = 4𝑥 − 1 algebraicially

Example 9 – Solve |𝑥2 − 7𝑥 + 2| = 10

Example 10 – Solve |𝑥 − 2| = 𝑥 − 2

Check:

Check:

quadratic absolute value equations

4.3A – Absolute Value Functions Part 1

Example 1 – Graph 𝑦 = |𝑥|

In general, for absolute value functions: 𝑦 = |𝑓(𝑥)| 𝑖𝑠

Graphing Absolute Value Equations

To graph 𝒚 = 𝒂|𝒃𝒙 + 𝒄| + 𝒅 :

1) Find the x-coordinate of the vertex by solving 𝑏𝑥 + 𝑐 = 0. The vertex is (−𝑐

𝑏, 𝑑).

2) Construct a table of values, using values to the left & right of the x-coordinate of the vertex.

3) Plot the points. The graph is symmetrical about the vertex, and opens up if 𝑎 > 0, down if 𝑎 < 0.

Example 2 – Graph 𝑦 = −1

2|3 − 𝑥| + 5. State the domain, range, intercepts, and write as

a piecewise function. Step 1:

x y

x y

-3

-2

-1

0

1

2

3

The graph 𝑦 = |𝑥|

consists of two graphs:

This is why an absolute

value graph is called a

piecewise function.

Domain, Range, intercepts, piecewise function:

Example 3 – Graph 𝑦 = 2|2𝑥 − 1| − 4. State the domain, range, intercepts, and write as a

piecewise function.

4.3B – Absolute Value Functions Part 2

At times, absolute value functions are set up analogous to quadratic functions in standard

form, using h, k, and a values to graph the function.

Look back at last day’s notes at the graph of 𝑦 = |𝑥|. What is the basic count?

Summarize how each component of the function affects the graph:

𝑦 = ±𝑎|𝑥 − ℎ| + 𝑘

Example 1 – Graph each function

a) 𝑦 = |𝑥| − 3 b) 𝑦 = |𝑥 + 4|

c) 𝑦 = −2|𝑥| d) 𝑦 = −1

2|𝑥 + 1| + 2

e) 𝑦 = 3|𝑥| − 7 f) 𝑦 = 3 −1

4|𝑥 − 2|

Example 2 – For e) and f) , state the domain, range, intercepts, & write as piecewise functions:

If a graph is constructed for any function 𝑓(𝑥), what will the graph for |𝑓(𝑥)| look like?

Example 3 – Graph 𝑦 = 𝑥2 − 4 using a table of values. Then graph 𝑦 = |𝑥2 − 4|

𝑥 𝑓(𝑥) 𝑥 |𝑓(𝑥)|

Example 4 – Given each graph of 𝑦 = 𝑓(𝑥), graph 𝑦 = |𝑓(𝑥)|.

4.5A – Rational Functions Part 1

A function 𝑓 is a rational function if 𝑓(𝑥) =𝑔(𝑥)

ℎ(𝑥), where 𝑔(𝑥) and ℎ(𝑥) are polynomials.

The domain of 𝑓 consists of all real numbers except 𝑥 values that make the

denominator equal to zero (undefined values).

Example 1 – What are the undefined values for each rational function?

a) 2

𝑥+5 b)

𝑥

𝑥2−4 c)

𝑥+4

𝑥2−2𝑥−15 d)

𝑥−7

𝑥2+16

Graphing Rational Functions

Example 2 – Graph 𝑦 =1

𝑥

An asymptote is not part of the graph. A vertical asymptote is a line the graph

approaches as the denominator approaches zero. A horizontal asymptote Is a line the

graph approaches as |𝑥| gets larger. Not every rational function has both a horizontal

AND vertical asymptote.

Vertical Asymptotes are found when the denominator equals zero. Sometimes the

denominator must be factored to find the vertical asymptotes (see Example 1 above).

𝑥 𝑦

1

2

3

10

100

1000

0

0.5

0.2

0.1

0.01

0.001

-1

-2

-10

-100

-0.5

-0.1

-0.01

-0.001

The function is not defined for 𝑥 = 0, so this is a vertical asymptote of the function, and is drawn as a dashed line.

The pattern in the table can be written as: as 𝑥 → 0+ , 𝑓(𝑥) → +∞ as 𝑥 → 0− , 𝑓(𝑥) → −∞

as 𝑥 → +∞ , 𝑓(𝑥) → 0+ as 𝑥 → −∞ , 𝑓(𝑥) → 0− The line defined by 𝑦 = 0 is said to be a horizontal asymptote of the function, and is drawn as a dashed line.

Example 3 – Graph 𝑦 =1

𝑥−1− 1

What is the vertical asymptote? Plot it.

Now, complete the table of values below:

Another way to think about the function above is using 𝑦 =1

𝑥−ℎ+ 𝑘, where the

intersection of the two asymptotes is (ℎ, 𝑘). So, 𝑦 =1

𝑥−1− 1 is just like 𝑦 =

1

𝑥 , but

shifted one unit right and one unit down.

Before we graphed 𝑦 =1

𝑥−1− 1 above, we knew the vertical asymptote, but the

horizontal asymptote wasn’t apparent until after. How can we find the horizontal asymptote before? The horizontal asymptote is the value that 𝑦 approaches as 𝑥 approaches ±∞. Consider

the function 𝑓(𝑥) =𝑔(𝑥)

ℎ(𝑥) :

1) If ℎ(𝑥) is a higher power than 𝑔(𝑥), the horizontal asymptote is 𝑦 = 𝑘. 2) If 𝑔(𝑥) and ℎ(𝑥) have the same power, the horizontal asymptote is

𝑦 =𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟

𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

3) If 𝑔(𝑥) is a higher power than ℎ(𝑥), there is no horizontal asymptote.

Let’s investigate all three cases:

a) 𝑓(𝑥) =1

𝑥−2 b) 𝑓(𝑥) =

1

𝑥−2+ 3 c) 𝑓(𝑥) =

2𝑥

𝑥2−4 d)

3𝑥+1

𝑥−2 e) 𝑦 =

3𝑥(2𝑥−1)

4𝑥2−1 f) 𝑦 =

𝑥2

𝑥

𝑥 𝑦

2

0

1-

1+

-1000

1000

4.5B – Rational Functions Part 2

Example 1 – Graph 𝑓(𝑥) =1

𝑥+3

*To find the shape of the graph, it’s helpful to choose 𝒙 values one less and one greater

than vertical asymptote(s), and also 𝒙 values approaching both horizontal and vertical

asymptotes.

How would the graph be different if the function was 𝑔(𝑥) =1

𝑥+3− 4 ?

Example 2 – Graph 𝑦 =2𝑥

𝑥−1

𝑥 𝑦

𝑥 𝑦

Example 3 – Graph ℎ(𝑥) =2𝑥2

𝑥2+1

Example 4 – Graph 𝑓(𝑥) =2

𝑥2−𝑥−2

Example 5 – Graph 𝑦 =𝑥2−3𝑥−4

𝑥2+2𝑥

*A note about horizontal asymptotes:

𝑥 𝑦

𝑥 𝑦

𝑥 𝑦

4.6 – Reciprocal Functions

What is a reciprocal?

The following points are on the vertical number line: 4, 3, 2, 1

Plot and label their reciprocals:

The reciprocal of 1 is _____.

As the numbers increase towards infinity, their reciprocals…

For negative numbers:

The reciprocal of -1 is ______.

As the numbers increase towards negative infinity, their

reciprocals…

What is the reciprocal of 1

100? What is the reciprocal of

1

100 000?

As numbers decrease toward zero, how do their reciprocals behave?

As negative numbers increase toward zero, how do their reciprocals behave?

4

3

2

1

0

Example 1 – Graph 𝑦 = 𝑥 and its reciprocal on the same coordinate plane

Notice that the x-intercept on the original function becomes the vertical asymptote on the

reciprocal function. Remember, the x-intercept is where 𝑦, or 𝑓(𝑥) is equal to zero, so when

this becomes a reciprocal, 1

𝑓(𝑥), the reciprocal function is now undefined.

𝑥 𝑦 𝑦 =1

𝑥 𝑥 𝑦 𝑦 =

1

𝑥

-10

-5

-2

-1

−1

2

−1

5

−1

10

0

1

10

1

5

1

2

1

2

5

10

What is unique

about the

reciprocals of -1

and 1 in this

example?

In this example, (1,

1) and (-1, -1) are

called invariant

points, as they are

the same for the

original and

reciprocal.

Invariant points

are always where

𝑓(𝑥) = ±1

Before plotting points on the graph, let’s find any vertical and horizontal

asymptotes for 𝑦 =1

𝑥

using methods learned in section 4.5A.

Vertical Asymptote:

Horizontal Asymptote:

Example 2 – Graph 𝑦 = 𝑥2 − 4 and then graph its reciprocal by first identifying any

asymptotes, and then identifying invariant points.

Example 3 – Draw the graph of the reciprocal function using the original function graph.