### Key Concepts

### 3.1 Graph Linear Equations in Two Variables

**Points on the Axes**- Points with a
*y*-coordinate equal to 0 are on the*x*-axis, and have coordinates $\left(a,0\right).$ - Points with an
*x*-coordinate equal to $0$ are on the*y*-axis, and have coordinates $\left(0,b\right).$

- Points with a
**Quadrant**$$\begin{array}{cccc}\hfill \mathbf{\text{Quadrant I}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant II}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant III}}\hfill & \hfill \phantom{\rule{2em}{0ex}}\mathbf{\text{Quadrant IV}}\hfill \\ \hfill (x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill & \hfill \phantom{\rule{2em}{0ex}}(x,y)\hfill \\ \hfill (+,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,+)\hfill & \hfill \phantom{\rule{2em}{0ex}}(-,-)\hfill & \hfill \phantom{\rule{2em}{0ex}}(+,-)\hfill \end{array}$$**Graph of a Linear Equation:**The graph of a linear equation $Ax+By=C$ is a straight line.

Every point on the line is a solution of the equation.

Every solution of this equation is a point on this line.**How to graph a linear equation by plotting points.**- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

*x*-intercept and*y*-intercept of a Line- The
*x*-intercept is the point $\left(a,0\right)$ where the line crosses the*x*-axis. - The
*y*-intercept is the point $\left(0,b\right)$ where the line crosses the*y*-axis.

- The
**Find the***x*- and*y*-intercepts from the Equation of a Line- Use the equation of the line. To find:

the*x*-intercept of the line, let $y=0$ and solve for*x*.

the*y*-intercept of the line, let $x=0$ and solve for*y*.

- Use the equation of the line. To find:
**How to graph a linear equation using the intercepts.**- Step 1. Find the
*x*- and*y*-intercepts of the line.

Let $y=0$ and solve for*x.*

Let $x=0$ and solve for*y*. - Step 2. Find a third solution to the equation.
- Step 3. Plot the three points and check that they line up.
- Step 4. Draw the line

- Step 1. Find the

### 3.2 Slope of a Line

**Slope of a Line**- The slope of a line is $m=\frac{\text{rise}}{\text{run}}.$
- The rise measures the vertical change and the run measures the horizontal change.

**How to find the slope of a line from its graph using**$m=\frac{\text{rise}}{\text{run}}.$- Step 1. Locate two points on the line whose coordinates are integers.
- Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
- Step 3. Count the rise and the run on the legs of the triangle.
- Step 4. Take the ratio of rise to run to find the slope: $m=\frac{\text{rise}}{\text{run}}.$

**Slope of a line between two points.**- The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is:

$$m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}.$$

- The slope of the line between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is:
**How to graph a line given a point and the slope.**- Step 1. Plot the given point.
- Step 2. Use the slope formula $m=\frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
- Step 3. Starting at the given point, count out the rise and run to mark the second point.
- Step 4. Connect the points with a line.

**Slope Intercept Form of an Equation of a Line**- The slope–intercept form of an equation of a line with slope
*m*and*y*-intercept, $\left(0,b\right)$ is $y=mx+b$

- The slope–intercept form of an equation of a line with slope
**Parallel Lines**- Parallel lines are lines in the same plane that do not intersect.

Parallel lines have the same slope and different*y*-intercepts.

If ${m}_{1}$ and ${m}_{2}$ are the slopes of two parallel lines then ${m}_{1}={m}_{2}.$

Parallel vertical lines have different*x*-intercepts.

- Parallel lines are lines in the same plane that do not intersect.
**Perpendicular Lines**- Perpendicular lines are lines in the same plane that form a right angle.
- If ${m}_{1}$ and ${m}_{2}$ are the slopes of two perpendicular lines, then:

their slopes are negative reciprocals of each other, ${m}_{1}=-\frac{1}{{m}_{2}}.$

the product of their slopes is $\mathrm{-1},$${m}_{1}\xb7{m}_{2}=\mathrm{-1}.$ - A vertical line and a horizontal line are always perpendicular to each other.

### 3.3 Find the Equation of a Line

**How to find an equation of a line given the slope and a point.**- Step 1. Identify the slope.
- Step 2. Identify the point.
- Step 3. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4. Write the equation in slope-intercept form.

**How to find an equation of a line given two points.**- Step 1. Find the slope using the given points. $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
- Step 2. Choose one point.
- Step 3. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 4. Write the equation in slope-intercept form.

To Write an Equation of a Line **If given:****Use:****Form:**Slope and *y*-intercept**slope-intercept**$y=mx+b$ Slope and a point **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$ Two points **point-slope**$y-{y}_{1}=m\left(x-{x}_{1}\right)$

**How to find an equation of a line parallel to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the parallel line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form: $y-{y}_{1}=m\left(x-{x}_{1}\right).$
- Step 5. Write the equation in slope-intercept form

**How to find an equation of a line perpendicular to a given line.**- Step 1. Find the slope of the given line.
- Step 2. Find the slope of the perpendicular line.
- Step 3. Identify the point.
- Step 4. Substitute the values into the point-slope form, $y-{y}_{1}=m\left(x-{x}_{1}\right)$
- Step 5. Write the equation in slope-intercept form.

### 3.4 Graph Linear Inequalities in Two Variables

**How to graph a linear inequality in two variables.**- Step 1. Identify and graph the boundary line.

If the inequality is $\le \text{or}\ge ,$ the boundary line is solid.

If the inequality is $<\phantom{\rule{0.2em}{0ex}}\text{or}>,$ the boundary line is dashed. - Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
- Step 3. Shade in one side of the boundary line.

If the test point is a solution, shade in the side that includes the point.

If the test point is not a solution, shade in the opposite side.

- Step 1. Identify and graph the boundary line.

### 3.5 Relations and Functions

**Function Notation:**For the function $y=f(x)$*f*is the name of the function*x*is the domain value- $f(x)$ is the range value
*y*corresponding to the value*x*

We read $f(x)$ as*f*of*x*or the value of*f*at*x*.

**Independent and Dependent Variables:**For the function $y=f(x),$*x*is the independent variable as it can be any value in the domain*y*is the dependent variable as its value depends on*x*

### 3.6 Graphs of Functions

**Vertical Line Test**- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.

**Graph of a Function**- The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$

$$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$$

- The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$
**Linear Function**

**Constant Function**

**Identity Function**

**Square Function**

**Cube Function**

**Square Root Function**

**Absolute Value Function**