Unit 7 Notes: Post Keystone Exam Topics

Algebra 1 Mrs. Bondi

2315 extra function inequalitiesAlgebra 1

Unit 7: Post Keystone Exam Topics

Lesson 1 (PH Text 4.2, 5.8, 9.1, 7.6, 10.5): Families of Functions

Lesson 2 (PH Text 9.3&4): Solving Quadratic Equations

Lesson 3 (PH Text 9.6): Solving Quadratic Equations using the Quadratic Formula

Lesson 4 (PH Text 5.8): Solving Absolute Value Equations

Lesson 5 (PH Text 2.5): Solving Literal Equations

Lesson 6 (PH Text 10.4): Solving Radical Equations

Lesson 7 (PH Text p.605): Midpoint and Distance Formulas

Lesson 8 (6.0 only) (PH Text 10.2-3): Operations with radicals involving variables

Lesson 9 (6.0 only) (PH Text 10.2-3): Division of Radicals involving conjugates

Lesson 1 (PH Text 4.2, 5.8, 9.1, 7.6, 10.5): Families of Functions

Objective: to identify families of functions for equations and graphs

to predict what the graph of an equation will look like

Explore - Using a Graphing Calculator:

1)Graph each function. Sketch a picture of each graph.

Explore more on your own…

Summarize:

2)Sort the graphs into sixcategories by grouping how they look.

3)What similarities among the graphs do you see in each category?

4)What differences do you see?

These six Families of Functions are:

Type / Linear / Quadratic / Cubic
Graph
Equation
Type / Absolute Value / Radical / Exponential
Graph
Equation

Families of Functions:

Linear function -

highest power of x is ______

graph forms a ______

(when looking from left to right)

positive coefficient of x, line slants ______

negative coefficient of x, line slants ______

Quadratic function -

highest power of x is ______

graph forms a ______, or a ______

positive coefficient of x, graph opens ______

negative coefficient of x, graph opens ______

Cubic function -

highest power of x is ______

graph forms a ______

(when looking from left to right)

positive coefficient of x, graph curves ______

negative coefficient of x, graph curves ______

Absolute Value function -

has an absolute value symbol around a variable expression

graph forms a ______

positive coefficient of x, graph opens ______

negative coefficient of x, graph opens ______

Radical function -

highest power of x is ______

graph forms a ______

positive coefficient of x, graphgoes ______

negative coefficient of x, graphgoes ______

Exponential function -

In the form y = a · bx, where a ≠ 0, b > 0, b ≠ 1,

and x is a real number

graph forms a ______

multiplier (b) is >1, graph opens ______

multiplier (b) is <1, graph opens ______

positive constant (a), graph opens ______

negative constant (a), graph opens ______

Class Practice:

Sketch an example of each of the following:

1.)Quadratic2) Cubic 3) Radical

4) Absolute Value5) Linear 6) Exponential

Identify the Function Family to which each belongs. Describe how you knew:

4) 5) 6)

7) y = 6x2 + 18) y = 4x – 19) y = x2 + 3x + 2

10) y = 3x11) 12) y = 7 – x3

Lesson 2 (PH Text 9.3 & 4): Solving Quadratic Equations

Objectives: to solve quadratic equations by graphing , using square roots, and factoring

A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0, where a ≠ 0.

The standard form of a quadratic equation is ax2 + bx + c = 0.

Quadratic equations can be solved in a variety of ways. We will consider graphing, using square roots, and factoring.

Solving quadratic equations by graphing:

Graph the function. The x-values of points where the graph crosses the x-axis are considered the solutions.

Solving quadratic equations by using square root:

Isolate the squared term. Find the square roots of each side, and simplify.

2x2 – 98 = 0

Finding solutions to quadratic equations is also an important use of factoring. A replacement set is the set of all solutions to a polynomial equation. Any real numbers that make an equation a true statement are a part of the equation’s replacement set.

The easiest way to find an equation’s solution set is to apply the zero-product property.

Zero-Product Property:

For all real numbers a and b, if ab = 0 then a = 0, b = 0 or both a and b = 0.

To find the replacement set to a given equation:

Write the equation in standard form.
Factor the polynomial completely.
Set each of the factors equal to zero.
Solve for the variable.
Check by replacing the potential solution for the variable in the equation. If it results in a true statement, it is a part of the equation’s replacement set.

Example:

8x2 + 10x –3 factored is(2x + 3) (4x – 1)so (2x + 3) = 0 and/or (4x – 1) = 0

Check:Replacement set for 8x2 + 10x –3: { }

Find the possible solutions for 4x2 – 21x = 18

HW: p.551 #24-36 even; p.558#18-36 even

Lesson 3 (PH Text 9.4): Solving Quadratic Equations using the Quadratic Formula

Objectives: to solve quadratic equations by using the quadratic formula

HW: p.571 #12-28 even, 42

Lesson 4 (PH Text 3.7): Solving Absolute Value Equations

Objective: to solve equations that involve absolute value

Some equations may have two solutions. One time this can happen is when the variable is within absolute value bars.

two solutionsone solutionno solution

Examples:

x = 5 or -5x = 5absolute value can

never be negative

Class Practice:

1)2)3)

4)5)6)

7)8)

9)Jeff estimates his stride is 16 inches. However, any given stride is likely to vary from this estimate by up to 2 inches. Write and solve an equation to find Jeff’s minimum and maximum stride length.

HW: p.211 #10-30 even
Lesson 5 (PH Text 2.5): Solving Literal Equations

Objective: to use the strategies of reciprocals and opposites to solve equations involving variables only

A literal equation is an equation that expresses a relationship among variables.

Formula - shows the relationship between two or more variables

You can transform (change) a formula to define a different variable by “solving for” that variable. Use the skills we learned to solve equations.

Example: I = prt

Class Practice: Solve for the underlined variable. Show your work!

1) 2)

5) 6)

Solve each for the given variable.

7)for w8) for b9)d = rt for t

Lesson 6 (PH Text 10.4): Solving Radical Equations

Objective:to solve equations containing radicals;

to identify extraneoussolutions.

For any real number n,

Remember:

Simplify.

a) b) c)

Example: Simplify.

a) b) c)

Class Practice:

Simplify.

1) 2)

3) 4)

Example 2: Evaluate a) for n = 3b) for n= 9

Class Practice:

Evaluate for the given variable. Then simplify, if possible.

5) 6)

Example 3: For what values of y will be a real number?

Class Practice:

Find all values of x that make each radical expression a real number.

7) 8)

9) 10)

My Class Practice:

Find all values of x that make each radical expression a real number.

1)2)

3)4)

5)6)

7)8)

Lesson 7a (PH Text p.605): Distance Formula

Objectives: To find the distance between two points in a coordinate plane;

1. Graph the points ( –3, 4 ), ( 1, 1 ), ( –3, 1) and connect them

to form a triangle. Mark the lengths of the legs by counting units.

Use the Pythagorean Theorem to Now use the distance formula to find

find the length of the hypotenuse.the length between (1,1) and ( –3, 4 ).

d =

The distance formula: For points and in the coordinate plane, the distance d between the points is given by:

d =Round answers to the nearest tenth!!!!

2. Find the distance between (1, 4) and (−2, −5).3. Find the distance between (−3, 2) and (3, −2).

4. One hiker is 4 miles west and 3 miles north of the campground. Another is 6 miles east and 3 mile

south of the campground. How far apart are the hikers? (the camp ground is at (0, 0) )

5. Mickey travels 15 miles west, then 20 miles north. Jamie travels

5 miles east, then 10 miles south. How far apart are they?

6. Quadrilateral KLMN has vertices with coordinates K(-3, -2), L(-5, 6), M(2, 6) and N(4, -2).

a. Show that .

b. Use slopes to show that and are parallel.

HW: p.605 #1-6 (find the distance between the points only)

Lesson 7b (PH Text p.605): Midpoint Formula

Objectives: To find themidpoint of two points.

1. Graph the points ( –3, 4 ), ( 1, 2 )

Find the point that is half-way between the two given points,

the midpoint.

The midpoint of a segment is the halfway point between two endpoints. The coordinates of a midpoint are the averages of the coordinates of the endpoints.

The midpoint formula: For endpoints and on the coordinate plane the midpoint m can be expressed by:

M =

Find the midpoint of A(2, -1) and B(4, -3) Find the midpoint of P(−1, 6) and Q(5, 0)

To find a second endpoint when you are given one endpoint and the midpoint…

Use:

Midpoint (1,3)Midpoint formula M =

Given endpoint (2,-5)

Missing endpoint (x,y)

HW: p.605 #1-6 (find the midpoint)

The distance formula:

The midpoint formula:

Lesson 8 (6.0 only) (PH Text 10.2-3): Operations with radicals involving variables

Objective: to add, subtract, multiply and divide radicals involving variables.

To add and subtract, simplify first, then combine like terms.

Example: Simplify. Make sure all answers are in simplest radical form.

1) 2)

Remember when multiplying, multiply the coefficients (number outside) then multiply the radicands (the number inside). Simplify your answers if possible.

Example: Simplify. Make sure all answers are in simplest radical form.

3) 4)

To divide, simplify first using power rules, then rationalize the denominator if necessary.

Example: Simplify. Make sure all answers are in simplest radical form.

5) 6)

Class Practice: Perform the indicated operations. Express your answers in simplest radical form.

1) 2)

3) 4)

5) 6)

7 ) 8)

9) 10)

Lesson 9 (6.0 only) (PH Text 10.2-3): Division of Radicals involving conjugates

Objective: Be able to rationalize denominators using conjugates.

The radical expression is called the conjugate of . You use this to rationalize the denominator of a fraction when adding and subtracting is involved.

If m and n are non-negative, then the binomials and are conjugates.

The sum and difference of the same two terms.

Class Practice:

Write the conjugate of each binomial.

1) 2 ) 3)

Example: Rationalize the denominator.

a) b)

Class Practice:

Rationalize the denominator and simplify.

4) 5) 6)

7) 8) 9)

Section 4: Solving Radical Equations.

Objective: Be able to solve radical equations

A radical equation is an equation that contains radicals with variables in the radicand.

To solve:

1)Isolate the radical on one side of the equation and combine any like terms

2)Square both sides to eliminate the radical

3)Repeat steps 1 and 2 if necessary.

4)Check your answer.

Example: Solve and Check.

1) 2)

Class Practice: Solve and check.

1) 2)

3) 4)

5) 6)

7) 8)