Algebra 2 ­ Chapter 2
Domain ­ input values, X (x, y)
Range ­ output values, Y (x, y)
Function ­ For each input, there is exactly one output
Example:
Vertical Line Test ­ a relationship is a function, if NO vertical line intersects the graph more than
once
Function Notation: renaming y as f(x) Ex.
Slope intercept Form:
Slope information
• Slope formula: Y2 – Y1​
Rise
X2 – X1​
Run
• Parallel lines: have same slope
• Perpendicular lines: have opposite reciprocal slopes ex. 2, ­1/2
• Line with positive slope: increases from left to right
• Line with negative slope: decreases from left to right
• Horizontal line: 0 slope y = some #
• Vertical line: undefined slope x = some #
• Quadrants: I, II, III, IV – counterclockwise starting from real life (+) #s.
• Slope intercept form: y = mx + b
3 Ways to Graph a Line
1. Table of values / T chart
• Pick values for x to be
• Plug into formula
• Solve for y
• choose values that can fit into the equation easily, 0 and multiples of the variable term work
well.
• Graph the ordered pairs, connect points to form line.
2. Slope intercept form y = mx + b
• May need to get equation into this format by subtracting x from both sides, and then dividing by
the # in front of y.
• b tells where to start on y axis, m tells rise over run (slope).
• Plot points, connect dots to form line.
3. Use x or y intercepts
• Works well when equation is in standard form ax+ by = c
• Plug zero in for x, solve for y. ( 0 , ? )
• Plug zero in for y, solve for x. ( ? , 0 )
• Graph these two ordered pairs, connect dots to form line.
3 ways to write an equation
1. Given slope and y intercept
• Ex. Write the eq. of the line w/slope of 2/3 and y int. of 3.
• Plug info into y = mx + b ex. Y = 2/3x + 3.
• Start at 3 on y int, go up 2, over 3.
2. Given slope and a point
• Ex. Write eq. of line w/ slope of 3 that goes thru ( 2, 4)
• Use y = mx + b
• 1st determine what we know ex. M = 3, x = 2, y = 4.
• What do you need? Ex. B = ?
• Plug in m, x, y into y = mx + b Ex. 4 = 3(2) + b à 4 = 6 + b à ­2 = b
• Rewrite in y = mx + b. Ex. Y = 3x + ­2
3. Given 2 points
• Ex. Write the eq that passes thru ( 3, 6) and ( ­2, 5)
• 1st find the slope y2 – y1 5 – 6 = ­1 = 1
x2 – x1 ­2 – 3 ­5 5
• Use slope and one point ( you choose) and then use method #2.
• Ex. 6 = 1/5(3) + b
• Solve for b.
• Rewrite in y = mx + b.
Direct Variation
• Two variables have the same rate and ratio regardless of the values of the variables. K is a
constant variable in both.
• Ex. Y = K is the same as Y = KX.
X
• Write the eq for the story problem
• Solve for K
• Plug the answer for K back in to the equation you wrote to find answer
Rate of Change
• Determine if there is an increase ( + slope) or decrease ( ­ slope)
• Read the chart/table to determine years passed and change in profit/etc
• Determine slope rise/run à y axis info/ x axis info
• Ex. 16, 000 profit / 4 years passed
Graphing Inequalities
• Get into y = mx + b form if needed
• Graph line
• Use solid line for >, dashed line for >.
• Choose ( 0, 0 ) and plug into equation
• Shade in the direction of the “true” answers.
3 ways to graph Absolute Value
1. Make table of values. ( book method)
• The corner point is the vertex – where values start to go back up – v shaped
• Y = a | bx + c | + d
• Find the x coordinate by finding the value of x that makes | bx + c | = 0
• Ex. Y = ­ | x – 2 | + 3
• The x coordinate ( x, y ) for the vertex ( turning pt) is 2 , because x – 2 = 0.
• If x = 2, y = 3.
• Now construct table of values using 2 as x coordinate
• Choose #s that are ahead and behind 2 to plug into original eq.
• Plot these ordered pairs. You should get a v shaped line, if not, double check eq.
2. Can use calculator.( not sure of steps)
• Math key, number, abs, y1 = ( plug in eq)
• Zoom standard
• Second key, graph
• Scroll down to find vertex
• 2nd trace or calculate key – choose minimum, give x and y values to the left and right of lowest
pt on the graph to find vertex.
3. Fill in terms. ( Clark method)
• Y = a | bx + c | + d
• Determine what each letter means
• Ex. Y = | x – 5 | a = 1, b = 1, c = 5, d = 0
• Use vertex formula : ( c/b , d ) Ex. 5/1 , 0 ( 5, 0) is vertex.
• If a is (+), graph opens up. If a is ( ­), graph opens down.
• Find slope m = ab. Ex. 1 x 1 = 1
• Start at vertex, ( 5, 0) , then plot slope 1/1 in both directions left and right. Should form V.
Function Transformations:
Transformation ­ Changes a graphs size, shape, position or orientation.
Translation ­ shifts a graph horizontally and/or vertically. Does not change its size, shape or
orientation.
fx = y
● Movement in relation to the original graph
● Inside the ( ), move the opposite of what you think, changing X values ( + goes left, ­
goes right)
● Outside the ( ), move exactly what you think, changing Y values (+ goes up, ­ goes down
Ex.
f(x+2)­3 Left 2, down 3
f(x­4) +1 Right 4, up 1
2 * f(x+3) ­1 Left 3, y * 2, down 1 (do order of operations ­ start in (), multiply, then move up or
down)
y < ­1/2 |x­2| + 1 Move right 2, up 1, opens down, slope is ­1/2
pick a point, plug in (x,y) values and solve to determine where to shade