Chapter 1: Functions and Graphs – Linear and Quadratic Functions (Notes)

Mathematical Modeling

• Mathematical modeling is the process of using mathematics to solve real-world problems.
• Three steps for mathematical modeling:
  • Construct the mathematical model.
  • Solve the mathematical model.
  • Interpret the solution to the mathematical model.
• Each step should be done in the context of the real-world problem being modeled.

The Cycle of Mathematical Modeling

• In complex problems, the cycle may need to be repeated to obtain the required information about the real-world problem.
• Linear and quadratic functions can be used to construct mathematical models of real-world problems.

Linear Equations in Two Variables

Definition of a Linear Equation

• A linear equation in two variables is an equation that can be written in the standard form Ax + By = C
  • The values A, B, and C are constants.
  • A and B are not both 0.
  • x and y are variables.

Theorem: Graph of a Linear Equation in Two Variables

• The graph of any equation in the standard linear form Ax + By = C is a line.
  • When A = 0, the line is horizontal.
  • When B = 0, the line is vertical.
• Any line graphed in a Cartesian coordinate system is the graph of an equation of this form.

Slope of a Line

• (Content presented as part of slope discussion; see Geometric Interpretation of Slope below.)

Equations of a Line

• For a line passing through the points (a,0) and (0,b), a is called the x-intercept and b is called the y-intercept.
• It is common practice to refer to either a or (a,0) as the x-intercept, and either b or (0,b) as the y-intercept.

Example: Equation of a Line

• Given slope m = 3 and point (x1,y1) = (2,5). Substitute into the point-slope form:
  • Point-slope form: y - y1 = m(x - x1)
  • Substitution: y - 5 = 3(x - 2)
• Convert to slope-intercept form: y = 3x - 1
• Standard form: -3x + y = -1
• A line with slope 3 passes through the point (2,5). Find equations in various forms (point-slope, slope-intercept, standard).

Linear Functions and Modeling

Linear Function

• (Slide indicates introduction to linear function; connected to models.)

Example: Linear Function Modeling

• A student organization plans an event with a custom-printed shirt project:
  • Fixed cost for the art work: 175
  • Printing cost per shirt: 4.75
  • Selling price per shirt: 12.00
• Tasks:
  • Write a function that gives the total cost for printing the shirts based on the number of shirts printed.
  • Write a function that gives the revenue from selling these shirts.
  • Find the number of shirts to break even (round to the nearest whole number).

Cost and Revenue Functions

• Fixed cost: C_{fixed} = 175
• Variable cost per shirt: 4.75
• Cost function (for x shirts): C(x) = 175 + 4.75x
• Revenue per shirt: 12
• Revenue function (for x shirts): R(x) = 12x
• Break-even when C(x) = R(x).
• Solve: 175 + 4.75x = 12x \ \ x = 24 (rounded to nearest whole number)

Interval Notation

• Interval notation provides a convenient way to represent collections of numbers given as inequalities or on a number line.
• The endpoints of an interval are denoted as a and b with a < b.
• An interval is closed if it contains its endpoints; open if it does not.
• A table summarizes interval notation, inequality notation, and line graphs.

Endpoints and Interval Types

• Endpoints: a and b with a < b are called the endpoints.
• Closed interval: contains endpoints; Open interval: does not contain endpoints.

The Square Function

• A basic elementary function is the square function: h(x) = x^2.
• The graph of the square function is a parabola.

Quadratic Functions: Definition and Properties

Definition

• The domain of any quadratic function is the set of all real numbers: ext{Dom} = \,oldsymbol{R}.
• The range of a quadratic function is a proper subset of the real numbers; the exact range will be discussed later.

Quadratic Function Concepts

• An x-intercept of a function is also called a zero of the function.
• The x-intercepts (if they exist) can be found by solving the quadratic equation y = ax^2 + bx + c = 0 for x.
• The leading coefficient a must be non-zero.

Vertex Form of the Quadratic Function

• Vertex form: f(x) = a(x - h)^2 + k.
• The vertex can be found by completing the square, or by graphing calculator techniques, or by other analytic methods.
• The axis of symmetry is x = h.
• If a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum.
• Domain: (-\, o\,\infty, +\infty) (all real numbers).
• Range:
  • If a > 0, ext{Range} = [k, \infty)
  • If a < 0, ext{Range} = (-\infty, k]

Completing the Square: Vertex Form Example

Example: Use Completing the Square to Find the Vertex Form

• Given: f(x) = -3(x^2 - 2x) - 1
• Step 1: Add inside the parentheses to complete the square: -3(x^2 - 2x + 1) - 1 + 3
• Step 2: Simplify: -3(x - 1)^2 + 2
• Vertex: (1, 2)
• Since the leading coefficient is a = -3 < 0, the parabola opens downward.

Intercepts of a Quadratic Function

• The y-intercept is found by evaluating f(0): for the example, f(0) = -1, so the y-intercept is (0, -1).
• The x-intercepts are found by solving f(x) = 0 for x (the roots of the quadratic).

Quadratic Function Properties (General)

• For f(x) = a(x - h)^2 + k with a e 0:
  • If a > 0, the graph is a parabola opening upward; vertex is a minimum; range is [k, \, \infty).
  • If a < 0, the graph is a parabola opening downward; vertex is a maximum; range is (-\infty, k].
  • The vertex coordinates are (h,k) and the axis of symmetry is x = h.
  • Domain is the set of all real numbers; (-\infty, \infty).

Example: Solve a Quadratic Inequality

• Inequality: -x^2 + 5x + 3 > 0
• Since a = -1 < 0, the parabola opens downward, so the solution set is between the x-intercepts (where the graph is above the x-axis).
• Intercepts found (via calculator or solving): x \,\approx\, -0.541 \,\text{and}\, 5.541.
• Interval notation for the solution: [-0.541, \ 5.541].
• Note: The calculator process might show a tiny positive value at the second root; interpret as zero (use y = 0).

An Example of Modeling Using Quadratic Functions: Peach Orchard

• Problem setup (Macon, Georgia): 20 trees per acre; each tree yields 300 peaches; for each additional tree, peaches per tree drop by 10.
• Yield model: Yield = (peaches per tree) × (number of trees).
• Current yield: 300 \times 20 = 6000 peaches.
• If one more tree is planted: yield = (300 - 10) × (20 + 1) = 290 × 21 = 6090.
• If two more trees: yield = (300 - 2\cdot 10) × (20 + 2) = 280 × 22 = 6160.
• Let x be the number of additional trees. Then yield as a function of x is
  • Y(x) = (300 - 10x)(20 + x) = -10x^2 + 100x + 6000.
• The function is quadratic and opens downward (a = -10 < 0).
• The vertex will give the number of trees for maximum yield; the x-coordinate of the vertex is the number of additional trees to maximize yield; the y-coordinate is the maximum yield.
• Vertex form (via completing the square) yields Y(x) = -10(x - 5)^2 + 6250. The vertex is (5, 6250), so the maximum yield is 6250 peaches, obtained by planting 5 additional trees.
• Graphical interpretation: The maximum occurs at the vertex; parabola opens downward.

Break-Even Analysis: Analytical and Graphical Solutions

• Example setup: Revenue and cost functions for x million cameras (domain 1 < x < 15):
  • R(x) = x(94.8 - 5x)
  • C(x) = 156 + 19.7x
• Break-even points are solutions to R(x) = C(x) with the given domain.

Analytical Solution

• Solve: x(94.8 - 5x) = 156 + 19.7x
• Rearranged to standard quadratic form: -5x^2 + 75.1x - 156 = 0 (equivalently, 5x^2 - 75.1x + 156 = 0).
• Quadratic formula gives: x = 2.490 \text{ or } 12.530.
• Break-even occurs at these two production levels: approximately 2.490 million and 12.530 million cameras.

Graphical Solution

• Graph the cost and revenue functions on a calculator within the domain 1 < x < 15.
• Intersections of the two graphs give break-even points: approximately 2.490 and 12.530 (million cameras).

Linear Regression and Scatterplots

Linear Regression

• Linear regression is the process of analyzing numeric information to find a linear function that matches as closely as possible.
• The resulting linear function is called the best-fit line.
• Technology often used to: create a scatterplot and find a linear regression model.

Create a Scatterplot

• Calculator steps to create a scatterplot:
  • Input the data.
  • Set up the type of plot desired.
  • Graph the scatter plot in a suitable window.

Example: Create a Scatter Plot

• Data for tire pressure x (psi) and mileage y (thousand miles):
  • x: 28, 30, 32, 34, 36
  • y: 45, 52, 55, 51, 47
• The data relate tire air pressure to mileage for an automobile tire manufacturer.
• Use calculator technology to create a scatterplot for the data.

Creating a Scatter Plot: Data Entry and Plot Setup

• Data entered via statistics editor (Stat Edit):
  • Tire pressure into L1 (list 1)
  • Mileage into L2 (list 2)
• The calculator screen shows the result.
• Scatter plot setup requires choosing the type and data lists in the stat plot menu and placing data in the correct lists.
• It is suggested to delete any equations in the graphing (y =) screen before viewing the scatter plot.
• Use Zoom 9: ZoomStat to size the graph window.
• The resulting scatter plot is shown.

Create a Linear Regression Model

• If the scatter plot suggests a linear model is not the best, you can still generate a linear regression model for comparison.
• Process: Stat Calc 4: LinReg(ax + b) to obtain the regression menu; indicate data locations (Usually L1 for x and L2 for y). The regression function is stored in Y1.
• Access the settings: Vars, Y-vars, Function, 1: Y1.
• Calculate to obtain the regression line: y = mx + b (example result: y = 0.15x + 45.2).
• The r^2 value (coefficient of determination) gives a measure of goodness-of-fit; r^2 converted to a percentage indicates the approximate accuracy of the fit.
• Scatter plot vs regression line: In this example, the scatter plot shows the data are not very linear, but the regression line is the best linear approximation among options.

Quadratic Regression (Best-Fit Parabola)

• Quadratic regression analyzes data to find a quadratic function that best fits the data: the best-fit parabola stored in Y2.
• Process: Stat Calc 5: QuadReg(ax + b) with data in L1 (x) and L2 (y); store the quadratic regression function in Y2.
• Example results: The quadratic regression function is y = -0.52x^2 + 33.29x - 480.94. The r^2 value, when converted to a percentage, indicates the quadratic model fits the data with about 95% accuracy.
• Graphs: Scatter plot with both regression lines show the quadratic regression function fits better than the linear regression line for this data.