Algebra Summary

Section 4.1 – Coordinates and Scatter Plots

Coordinate Plane – formed by two real number lines that intersect at right angle (90 degrees)

Ordered Pair – each point in the coordinate plane that corresponds to a pair of real numbers (for practical purposes it equals (x,y))

To plot an ordered pair – locate each point in the coordinate plane that corresponds to a to the x axis (first number – horizontal) and he Y-axis (second number- vertical)

Section 4.2 – Graphing Linear Equations

Solution of an Equation – the two numbers that make an equation with two variables true

Graph of an Equation – the set of ordered pairs that make an equation with two variables true

Graphing a linear Equation
1. Rewrite the equation in function form
2. Choose values of x (input) and find the corresponding out put (y) and create an input-output table
3. Plot the points on the graph and draw the line through the points to find all solutions to the equation

Linear Equation Graphs
1. x = a when x is a constant, the graph is a vertical line
2. y = a when y is a constant, the graph is a horizontal line

Section 4.3 – Quick Graphs Using Intercepts

X – intercept – the point on a graphical solution to an equation when the graph crosses the x-axis (y=0)

To solve the x-intercept
set the equation for when x is a function of y
Solve for x when y=0
The resulting coordinate is (x,0)

Y – intercept – the point on a graphical solution to an equation when the graph crosses the y-axis (x=0)

To solve the y-intercept
set the equation for when y is a function of x
Solve for y when x=0 The resulting coordinate is (0,y)

After the two points (x,0), (0,y) are plotted, the line formed through the two points represents all of the solutions to the equation.

Section 4.4 – The Slope of a Line

Slope – a ratio of the vertical change in a line to the change in horizontal
1. rise over run
2. change in y dived by change in x
3. vertical change to horizontal change

To de termine slope
1. Identify the coordinates of two points on the line
2. Define the points P1 and P2 as (x₁,y₁) and (x₂,y₂)
3. Determine the change in vertical by subtraction (y₂ - y₁)
4. Determine the change in horizontal by subtraction (x₁- x₁)
5. Determine the slope by dividing (y₂ - y₁) by (x₂- x₁)

The slope of an uphill line (moving from left to right) is positive
The slope of a downhill line (moving from left to right) is negative
The slope of a horizontal line is 0
The slope of a vertical line is undefined

Slope Triangle– a triangular method of determining slope. Find two points on the line (x₁,y₁) and (x₂,y₂). Locate a third point (x₂,y₁) using the coordinates from the two on the line. Draw a right triangle using two points on the line and the third point. This forms the slope triangle. The vertical leg is the rise and the horizontal leg is the run.

Section 4.5 – Direct Variation

Two number vary directly if there is a non zero number k which makes the fol lowing true
y = kx
where the constant of variation is the value for k

Direct Variation – when two quantities x and y vary directly

Properties of Direct Variation
1. The graph of y = kx is a line through the origin (0,0)
2. The slope of the line for y = kx is k

Section 4.6 – Quick Graphs Using Slope-Intercept Form

Slope-Intercept Form– A linear equation written in the form y = mx + b. In this form, m is the slope of the line and b is the y-intercept.

To graph a line using slope-intercept form
1. Rewrite the equation so that it appears in slope intercept form (y = mx + b)
2. Locate the y-intercept (0,b)
3. Use the slope triangle to locate a second point on the line.
4. Graph the line through the two points

Parallel Lines– two lines are parallel if they do not intersect at any point

To determine if two lines are parallel
1. Rewrite the equations for both lines in slope-intercept form
2. (y_a = m_ax_a + b_a, y_b = m_bx_b + b_b)
3. The lines are parallel if m_a = m_b

Section 4.7 – Solving Linear Equations Using Graphs

To solve a linear equation graphically
1. Write the original equation
2. Rewrite the equation in the form ax + b = 0
3. Rewrite the equation in the function form y = ax + b
4. Graph the line for y = ax + b
5. The solution for the original equation is the x-intercept of the graph

Section 4.8 – Functions and Relations

Function – a rule that establishes a relationship between two quantities, the input and the output. Each input will have only one output. An output may have more than one input.

Relation – any set of ordered pairs, a relation is not necessarily a function
Vertical line test – a relation is a function of the horizontal axis variable (typically x) if and only if no vertical line passes through two or more points on the graph

Function notation – noted by f(x), is used to describe the relationship between output y and input x. Read as the value of f at x or f of x.

Graph of a Function – the set of all points (x,f(x)) where x is the domain of the function. The notation f(x) is substituted for y in the ordered pair but has the same value.