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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