Topic 9: The Coordinate Plane; A graph of an equation is

Hartfield – College Algebra (Version 2015 - © Thomas Hartfield) Unit TWO

Topic 9: The Coordinate Plane; Graphs of Equations of Two Variables Review the properties and terminology associated with the coordinate plane:

A graph of an equation is the set of all points which are solutions to the equation. For an equation of two variables (for convenience, we will default to x and y), a graph is the set of all (x, y) coordinates which satisfy the equation. Consider the solution sets of the following equations: 3 1x 2 4 3x x 5x y

Graphing allows us to “see” an infinite number of solutions.


Review Graphing Linear Equations Recall that a linear equation of two variables is an equation which can be expressed in the form: Ax + By = C, where A, B, and C are real

numbers and both A ≠ 0, B ≠ 0. Other important forms of linear equations include: Slope-intercept form: y = mx + b, m is the slope includes the point (0, b) Point-slope form: y – y1 = m(x – x1) m is the slope includes the point (x1, y1) Vertical line: x = a, a is a real number Horizontal line: y = b, b is a real number

Ex. Restate the equation in slope-intercept form, identify the slope, and sketch a graph of the line on the coordinate plane.

3x + 4y = 12


Topic 10: Distance and Midpoint The distance d across the coordinate plane from point 1 1,x y to point 2 2,x y is determined by

2 22 1 2 1 .d x x y y

Ex. 1a Find the distance between 3, 2 and

1,4 .

The midpoint M of a line segment connecting point 1 1,x y to point 2 2,x y is determined by

1 2 1 2

, , .2 2x y

x x y yM

Ex. 1b Find the midpoint between 3, 2 and 1,4 .


Ex. 2 Find the distance from A to B and the location of the midpoint between A & B.

2, 5 1,4A B


Topic 11: Intercepts and Symmetry Definition: An intercept is a point of a graph on

an axis. If you have x and y axes, the any points on these axes would be x-intercepts and y-intercepts, respectively. To find intercepts of an equation for one variable, set the other variable equal to zero and solve. Thus, for an equation involving x and y variables: To find x-intercepts, To find y-intercepts, set y = 0 and set x= 0 and solve for x solve for y x = h1, x = h2, x = h3, … y = k1, y = k 2, y = k 3, … (h1, 0), (h2, 0), (h3, 0), … (0, k1), (0, k2), (0, k3), …

Ex. 1a Find the intercepts of 212 9 36.x y


Definition: Symmetry is the property of having exact correspondence of form and configuatation when an image is reflected about an axis or rotated about a point.

Two primary forms of symmetry: Reflective Symmetry: Rotational Symmetry:

Symmetry with respect to the x-axis - reflective symmetry

For a graph A, replacing any y with -y will not change A. Symmetry with respect to the y-axis

- reflective symmetry For a graph B, replacing any x with -x will not change B. Symmetry with respect to the origin

- rotational symmetry For a graph C, replacing any (x, y) with (−x, −y) will not change C.


To test for symmetry, apply the definitions: To test for symmetry with respect to the x-axis:

Replace y with −y in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric wrt the x-axis.

To test for symmetry with respect to the y-axis:

Replace x with −x in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric wrt the y-axis.

To test for symmetry with respect to the origin:

Replace (x, y) with (−x, −y) in the equation and simplify. If the original equation is returned, then the equation is symmetric wrt the origin.

Ex. 1b Determine if the graph of 212 9 36x y displays any type of symmetry.


Ex. 2 Find any intercepts for the equation 22 4 8x y and determine if its graph

displays any type of symmetry.


Topic 12: Circles Are both of these figures circles? Definition: A circle is the set of all points

equidistant to a fixed point. The fixed point is called the center

and the common distance from the center to each point of the circle is called the radius.

General Equation of a Circle: 2 2 0, where 0Ax By Cx Dy E A B

Ex. 1 2 2 6 4 12 0x y x y

No explicit information about circle conveyed. Center-Radius Form of a Circle (also called Standard Form):

2 2 2x h y k r Form reveals the center (h, k) and the radius r.


Steps to rewriting a circle equation into center-radius form: 1. Group like-lettered variables together and move the

constant to the other side. 2. Complete the square in each set of parentheses. 3. Compensate for the values added into the

parentheses. 4. Factor the groups and simplify both sides.

Ex. 1 Rewrite the equation into center-radius form (standard form), identify the coordinates of the center and the length of the radius, and sketch a graph on the coordinate plane.

2 2 6 4 12 0x y x y


Ex. 2 Rewrite the equation into center-radius form (standard form), identify the coordinates of the center and the length of the radius, and sketch a graph on the coordinate plane.

2 2 4 8 4 0x y x y

Ex. 3a Rewrite the equation into center-radius form (standard form), identify the coordinates of the center and the length of the radius, and sketch a graph on the coordinate plane.

2 2 10 2 22 0x y x y


Ex. 3b Calculate the x-intercepts and y-intercepts of the circle.

2 2 10 2 22 0x y x y


Ex. 4 Find the general equation of a circle from information.


A circle centered in the first quadrant

with a radius of 4 that is tangential to both axes.



A circle with diameter endpoints of

(9, −3) & (−1, 3).


Topic 13: Variation In some situations, it is easy to define one or more independent variables that impact a dependent variable. There are three forms of variation scenarios: Direct Variation y kx

Inverse Variation kyx

Joint Variation y kxz

Our approach to variation problems will come in three steps: 1. Use the information given to write a “general”

variation equation. 2. Use given data to solve for the constant of

proportionality (also called the constant of variation). 3. Merge the “general” variation equation and the

constant of proportionality to make a “specific” variation equation and solve.

Reminder: All variation equations must include a

constant of proportionality k.


Write a general variation equation from each statement. Ex. 1 P is directly proportional to u. Ex. 2 M varies inversely as t. Ex. 3 h is inversely proportional to the product of a and b.

Write a general variation equation from each statement. Ex. 4 S varies directly as the product of the squares of r and θ. Ex. 5 A is proportional to the second power of t and inversely proportional to the cube of x.


Ex. 6 Write a general variation equation from each statement and then use the data to find the constant of proportionality.

t is jointly proportional to x and y and

inversely proportional to r. When x = 2, y = 3, and r = 12, t = 25.

Ex. 7 Write a general variation equation from each statement and then use the data to find the constant of proportionality.

ε is proportional to a and inversely

proportional to the square of b. When a = 54 and b = 3, ε = 2.


Applications of Variation Ex. 1 The intensity of illumination from a light,

I, varies inversely as the square of the distance, d, from the light. A particular lamp has an intensity of 1000 candles at 8 yards. What will be the intensity of the lamp at 20 yards?

Ex. 2 The pressure of a sample of gas, P, is directly proportional to the temperature, T, and inversely proportional to the volume, V. If 100 L of gas exerts a pressure of 33.2 kPa at 400 K, determine the pressure exerted by the gas if the temperature is raised to 500 K and the volume is reduced to 80 L.


Ex. 3 The maximum weight, M, a beam can support is jointly proportional to its width, w, and the square of its height, h, and inversely proportional to its length, l. A beam with dimensions as shown in the picture below at left can support 4800 lbs. If a beam made from the same type of wood has the dimensions as shown in the picture below at right, what is the maximum weight it can support?


Topic 14: Function Basics Definition: A function f is a rule that assigns

each element x in a set A to exactly one element, called f(x), in a set B.

Set A is called the domain. The domain of a function f is the collection of values which are acceptable inputs for f.

Set B is called the range. The range of a function f is the collection of values which are possible outputs for f.

The variable of the domain is an independent variable. The variable of the range is a dependent variable.

The variable of the range, as notated f(x), can be read as the value of f at x or as the image of x under f.

Forms of Expressing Functions Verbal: (Usually) emphasizes the rule.

“Square the unknown and then subtract 3” Symbolic: example given by symbolic

function notation (see next page) Graphical: covered at a later point in notes Tabular: (Usually) emphasizes values of

the independent and dependent variables

Diagrammatical: example given by arrow diagram (see next page)

1 -2 4 13 0 -3 -2 1


Visual Representations of Functions Machine Diagram: Arrow Diagram:

Symbolic Function Notation Identify the parts of the each function below:

Ex 1a 1 5 , 0g t t t Ex 1b 3H x

2 3, 0f x x x

independent variable variable of the domain

dependent variable variable of the range

name of the function

rule of the function

conditions of the function(may not always be present)


Verbal versus Symbolic We must be able to switch between symbolic & verbal forms.

Example 2 3, 0f x x x “Square the unknown and then subtract three, only using unknown values of zero or greater”

Often the verbal expression will not define a letter for the unknown. Since any letter will suffice in this role, the independent variable in symbolic notation is sometimes referred to as a “dummy variable.”

Convert each function stated verbally into symbolic notation.

Ex. 2a Multiply by six, then add four. Ex. 2b Subtract two, divide by three, then take the square root.

Evaluating versus Solving Definition: To solve an equation, you want to

find values of the variable(s) which make the equation true.

Definition: To evaluate a function, you need a

value for the independent variable and by applying the rule of the function, you find the value of the dependent variable associated with it.

Solve: 6 7 10x Evaluate: 2 4f x x x for x = -1


Observe that an ordered pair can be created by the pairwise association of the independent and dependent variables. This allows for a graphical representation of a function. Example 3 4f x x

Note: A function may be expressed in graphical form and not have a well-defined symbolic or verbal form.

Vertical Line Test The vertical line test is designed to help determine if a graph can represent a function. The test may be expressed in a variety of ways, including the following form: Vertical Line Test: For a graph to represent a

function, any vertical line drawn over the graph may only intersect the graph at most once.

Why does the vertical line test work?

Because vertical lines include all the points involving one x-value (or first variable value). If a vertical line intersects a graph more than once, that means that the first variable is pairwise-associated with more than one second variable, which contradicts the definition of a function.


More on Evaluating Functions Functions can be evaluated for numerical values, constant values represented by letters, variables and variable expressions, or some other combination thereof. Ex. 3 Evaluate the function at the given values

of x. 2 4f x x x when x = a – 2

Ex. 4 Evaluate the function at the given values of x.

2 4f x x x when x = x + h.


Topic 15: Average Rate of Change in a function & The Difference Quotient Definition: The average rate of change of a

function ( )y f x between x = a and x = b is

change in ( ) ( )avg rate of change = change in

y f b f ax b a

The average rate of change is equal to the slope of the secant line between x = a and x = b on the graph of f; that is, the line passes through , ( ) and , ( ) .a f a b f b

Ex. 1 Find the average rate of change of f over the given interval.

37 1,2f x x x


Ex. 2 Find the average rate of change of f over the given interval.

3 1,4g x x x

The concept of a derivative in calculus is based upon an algebraic concept called the difference

quotient:

.f x h f x

h

In algebra, we can associate the difference quotient with the idea of average rate of change. x represents the value of a while x + h represents b (and thus the the difference in b and a is h). The difference quotient allows you to quickly find an average rate of change for many different a and b. The process of finding a difference quotient involves finding f(x + h), setting up the overall expression, and then simplifying the expression.


Ex. 1 Find the difference quotient for 3 4.f x x

Ex. 2 Find the difference quotient for 2 4 .f x x x


Ex. 3 Find the difference quotient

for 1.1

xf xx


Topic 16: Graphical Analysis A function is said to be increasing over an interval [a, b] if for all x1 < x2 in the interval, f(x1) < f(x2). A function is said to be decreasing over an interval [a, b] if for all x1 < x2 in the interval, f(x1) > f(x2). Ex. 1 Identify and/or determine the properties

of the function represented in the graph.

Identify the domain of f. Identify the range of f. Identify the intercepts of f. Identify the intervals over which f is increasing. Identify the intervals over which f is decreasing. Evaluate f(2). Solve f(x) = 5.


Ex. 2 Identify and/or determine the properties of the function represented in the graph.

Identify the domain of f. Identify the range of f. Identify the intercepts of f. Identify the intervals over which f is increasing. Identify the intervals over which f is decreasing. Evaluate f(2). Solve f(x) = 2.

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Topic 9: The Coordinate Plane; A graph of an equation is