Lesson 1: Parent Functions and Transformations

• Function Families

  • Functions that belong to the same family share key characteristics

  • The parent function is the most basic in a family

  • Functions in the same family are transformations of their parent function

  • Interval notation uses symbols:

    • ( -∞, ∞): all real numbers

    • {y}: represents a constant

  • Parent Functions:

    • Constant

      • Rule: f(x) = 1

      • Domain: All real numbers, ( -∞, ∞)

      • Range: y = 1, {1}

    • Linear

      • Rule: f(x) = x

      • Domain: All real numbers, ( -∞, ∞)

      • Range: All real numbers, ( -∞, ∞)

    • Absolute Value

      • Rule: f(x) = |x|

      • Domain: All real numbers, ( -∞, ∞)

      • Range: y ≥ 0, [0, ∞)

    • Quadratic

      • Rule: f(x) = x^2

      • Domain: All real numbers, ( -∞, ∞)

      • Range: y ≥ 0, [0, ∞)

• Transformations

  • A transformation changes the size, shape, position, and orientation of a graph

  • A translation is a transformation that shifts a graph either horizontally and/or vertically but does not change its size, shape or orientation

  • A reflection flips a graph over a line of reflection

  • Multiplying all the y-coordinates by a factor other than one will create a vertical stretch or shrink; if the factor is greater than 1 it is a stretch; if it is less than 1 it is a shrink

Lesson 2: Transformations of Linear and Absolute Value Functions

• Horizontal and Vertical Translations

  • Horizontal Translations

    • The graph of y = f(x - h) is a horizontal translation of the graph of y = f(x), where h ≠ 0

    • Subtracting h from the inputs before evaluating the function shifts the graph left when h < 0 and right when h > 0

  • Vertical Translations

    • The graph of y = f(x) + k is a vertical translation of the graph of y = f(x), where k ≠ 0

    • Adding k to the outputs before evaluating the function shifts the graph down when k < 0 and up when k > 0

• Reflections in the Axes

  • Reflections in the x-Axis

    • The graph of y = -f(x) is a reflection in the x-axis of the graph of y = f(x)

    • Multiplying the outputs by -1 changes their signs

  • Reflections in the y-Axis

    • The graph of y = f(-x) is a reflection in the y-axis of the graph of y = f(x)

    • Multiplying the inputs by -1 changes their signs

• Vertical and Horizontal Stretches and Shrinks

  • Vertical Stretches and Shrinks

    • The graph of y = a ⋅ f(x) is a vertical stretch or shrink by a factor of a of the graph of y = f(x), where a > 0 and a ≠ 1

    • Multiplying the outputs by a stretches the graph vertically (away from the x-axis) when a > 1, and shrinks the graph vertically (toward the x-axis) when 0 < a < 1

  • Horizontal Stretches and Shrinks

    • The graph of y = f(ax) is a horizontal stretch or shrink by a factor of 1/a of the graph of y = f(x), where a > 0 and a ≠ 1

    • Multiplying the inputs by a before evaluating the function stretches the graph horizontally (away from the y-axis) when 0 < a < 1, and shrinks the graph horizontally (toward the y-axis) when a > 1

Lesson 3: Modeling with Linear Functions

• Linear Equations

  • Slope-Intercept Form

    • y = mx + b

    • Use when given slope m and y-intercept b

  • Point-Slope Form

    • y - y1 = m(x - x1)

    • Use when given slope m and a point (x1, y1)

  • When given points (x1, y1) and (x2, y2), use the slope formula to find slope m, then use point-slope form with either point

    • Slope Formula: (y2 - y1)/(x2 - x1)

• Finding a Line of Fit

  • Step 1: Create a scatter plot of the data

  • Step 2: Sketch the line that most closely appears to follow the trend given by the data points; there should be about as many points above the line as below it

  • Step 3: Choose 2 points on the line and estimate the coordinates of each point; these points do not have to be original data points

  • Step 4: Write an equation of the line that passes through the two points from Step 3; this equation is a model for the data

  • You can also use a table and find 2 points and calculate the line of best fit from there

  • The line of best fit is the line that lies as close as possible to all of the data points

  • The correlation coefficient, denoted as r, is a number from -1 to 1 that measures how well a line fits a set of data pairs

    • When r is near 1, the points lie close to a line with a positive slope

    • When r is near -1, the points lie close to a line with a negative slope

    • When r is 0, the points do not lie close to the line

Lesson 4: Solving Linear Systems

• Solutions of Systems

  • A linear equation in three variables x, y, and z is an equation of the form ax + by + cz = d, where a, b, and c are not all zero

  • A solution of such a system is an ordered triple (x, y, z) whose coordinates make each equation true

  • The graph of each system is a three-dimensional space; the graphs of three systems form a three-dimensional shape that shows the number of solutions for the systems

    • One Solution

      • The graphs intersect in a single point

    • Infinite Solutions

      • The planes intersect in a line; every point on the line is a solution

    • No Solution

      • There are no points in common with all three planes

• Solving a Three-Variable System

  • Step 1: Rewrite the linear system in three variables as a linear system in two variables using the substitution or elimination method

  • Step 2: Solve the new linear system for both of its variables

  • Step 3: Substitute the values found in Step 3 into one of the original equations and solve for the remaining variable

  • When you obtain a false equation, such as 0 = 1, in any of the steps, the system has no solution

  • When you do not obtain a false solution, but obtain an identity such as 0 = 0, the system has infinite solutions