### Learning Outcomes

- Identify the equation of a semicircle
- Define a piecewise function or graph from an absolute value function or graph

In the Definite Integral section, we will learn how to find the area under a curve using integrals. Here we will review the equation of a semicircle, a curve we will often find the area under. We will also review the mathematical definition of the absolute value function, a skill needed to evaluate definite integrals that contain such a function.

## Identify the Equation of a Semicircle

The **standard form of a circle **is [latex](x-h)^2+(y-k)^2=r^2[/latex] where [latex](h,k)[/latex] is the center and [latex]r[/latex] is the radius.

If we take the standard form equation of a circle and solve for y, we obtain:

[latex]y=\pm \sqrt{r^2-(x-h)^2}+k[/latex]

[latex]y= \sqrt{r^2-(x-h)^2}+k[/latex] is the upper half of the circle and [latex]y=- \sqrt{r^2-(x-h)^2}+k[/latex] is the lower half of the circle.

### Example: Identifying the Equation of a SemiCircle

Identify the center and radius of the semicircle and graph. [latex]y=\sqrt{9-x^2}[/latex]

### Example: Identifying the Equation of a SemiCircle

Identify the center and radius of the semicircle and graph. [latex]y=-\sqrt{16-(x-2)^2}+1[/latex]

### Try It

Identify the center and radius of the semicircle and graph. [latex]y=\sqrt{4-x^2}[/latex]

## Define the Absolute Value Function

### A General Note: Absolute Value Function

The absolute value function can be defined as a piecewise function

[latex]\lvert x \rvert= \begin{cases} x ,\ x \geq 0 \\ -x , x < 0 \end{cases} [/latex]

Therefore, when evaluating an absolute value expression such as [latex]\lvert -9 \rvert [/latex], the true way to evaluate is as follows: [latex]\lvert -9 \rvert= -(-9) = 9 [/latex]

Now, when a variable is involved in the absolute value expression such as [latex]\lvert x-3 \rvert [/latex], notice what is inside the absolute value goes from being positive to negative at an *x*-value of 3. The piecewise breakdown of this absolute value function is as follows: [latex]\lvert x-3 \rvert= \begin{cases} x-3 ,\ x \geq 3 \\ -(x-3) , x < 3 \end{cases} [/latex]

### Example: Defining the Absolute Value Function

Give the piecewise breakdown of the absolute function [latex]\lvert x-4 \rvert [/latex].

### Try It

Give the piecewise breakdown of the absolute function [latex]\lvert 2x-3 \rvert [/latex].