GH

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