Chapter 1: Linear Functions

• 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

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