GOAL 4
The learner will use relations and functions to solve problems.
4.01 Use linear functions or inequalities to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret constants and coefficients in the context of the problem.
4.02 Graph, factor, and evaluate quadratic functions to solve problems.
4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.
4.04 Graph and evaluate exponential functions to solve problems.
4.01 Use linear functions or inequalities to model and solve problems; justify results.
Skill
Process for Solving Linear Equations
1. Identify the ______
2. Distribute.
3. Combine ______.
4. All ______on one side.
5. Add/Subtract to ______the ______.
6. ______all fractions.
Apply
The cost of renting a cab is $3.00 plus twenty-five cents per mile.
A. Define your variables.
B. Write an equation to represent the cost (C) of renting a cab.
C. Substitute 25 in for ______.
D. Substitute _____ in for C and solve for m.
Practice
1. Solve each equation for n or substitute -2 into each equation until you find one that is true.
2. Follow the process above and solve the equation. Do not forget to distribute all values in the parenthesis and combine like terms.
3. Use the key words to set up an equation. Check to make sure you know what you are solving for.
Model the Process
Solve the equation for the variable.
Step / Justify1. / 1.
2. / 2. Combine like terms. (7x and -2x)
3. / 3.
4. / 4. Add 6.
5. / 5.
Apply
A. C → ______
m → ______
B. C = ______m + ______
C. What would a twenty-five mile trip cost?
D. Suppose you were charged $12.75. How many miles did you ride?
Practice
1. Which equation has a solution of -2?
A. 4n + 3 = 11 B. 4 = 3n – 2
C. 5(1 + n) = -5 D. 3(n + 1) = 2
2. Solve 2(b – 3) + 5 = 3(b – 1).
A. -2 B. 2
C. -3 D. 3
3. A telephone company offers two long distance calling plans. Plan A charges 0.10 per minute and Plan B charges 0.07 per minute plus a monthly fee of $3.95. When, in terms of minutes, are the two calling plans equivalent?
4.01 Use linear functions or inequalities to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
Skill
Writing the equation of a line
1. Find the slope.
2. Use the slope and one point to write the equation of the line using the:
A. slope-intercept form:
B. point-slope form:
C. ratio:
(Note: If m is not a fraction, make it a fraction.)
OR
Using the graphing calculator
1. Press (STAT), (ENTER).
2. Enter the data in L1 & L2.
3. Press (STAT) → CALC→ 4:LinReg(ax + b), (ENTER)
4. Write the equation.
Apply
In 1980, the average price of a home in Greensboro was $40,000. By 2002, the average price of a home was $120,000.
A. Create a linear model based on this data.
B. Interpret the slope.
C. Predict using your equation. Remember is the years.
Practice
1. Find the value of the slope.
2. Predict using a model. The graphing calculator is good for this problem.
Model the Process (No calculator)
Write the equation of the line passing through the points (-2, 3) and (-4, 5).
Slope → ______
Equation →
Solve for y.
Now use the calculator to confirm your answer.
L1 / L2-2 / 3
-4 / 5
Press (STAT), (ENTER) →
a = ______
b = ______
Apply
A. y = ______x + ______
B. How much does the cost of a home change annually?
C. Estimate the price of a home in:
a. 1991: ______
b. 2006: ______
Practice
1. Denisha bought a car for $15,000 and its value depreciated linearly. After 3 years the value was $11,250. What is the amount of yearly depreciation?
A. $2000 B. $1,500
C. $1,250 D. $750
2. In 1994, the average price of a new domestic car was $16,930. In 2002, the average price was $19,126. Based on a linear model, what is the predicted average price for 2008?
A. $22,969 B. $21,322
C. $20,773 D. $18,577
4.01 Use linear functions or inequalities to model and solve problems; justify results.
b) Interpret constants and coefficients in the context of the problem.
Skill
Interpreting
1. y → Dependent variable.
2. x → Independent variable.
3. m → Rate of change.
4. b → Initial value.
Apply
For the line, where a > 0 and b > 0:
A. The y-intercept gives the value when x = 0. Look at where the line is on the x-axis as you move the line up.
B. The slope gives the rate at which the line is increasing or decreasing. Look at where the line adjusts on the x-axis as you increase the steepness of the line.
C. Vertical shift in the line.
D. A change in steepness of the line.
Practice
1. Write an equation in slope-intercept form. Then double the slope and y-intercept and look at the value of the x-intercept.
2. Determine whether the slope and y-intercept were increased or decreased.
3. Interpret the slope of the line. For each value of x, y increases or decreases by the value of the slope.
Model the Process
The equation below represents the charge for cell phone usage on a particular company plan.
C = .10m + 3.95
Define each value.
Variable/Value / MeaningCost
m
Cost per minute
3.95
Apply
A. If b increases and a remains constant, how does the x-intercept change?
B. If a increases and b remains constant, how does the x-intercept change?
C. If b is multiplied by -1 and a remains constant, how does the line change?
D. If a decreases, getting closer to 0, and b remains constant what happens to the line?
Practice
1. If the graph of a line has a positive slope and a negative y-intercept, what will happen to the x-intercept if the slope and the y-intercept are doubled?
A. The x-intercept becomes four times larger.
B. The x-intercept becomes twice as large.
C. The x-intercept becomes one-fourth as large.
D. The x-intercept remains the same.
2. If the slope of a line changes from -4 to -1/4 and the y-intercept changes from -2 to 0, then the graph of the line will be affected in what ways?
A. Less steep; up 2 units
B. Less steep; down 2 units
C. Steeper; up 2 units
D. Steeper; down 2 units
3. In the equation, if an x-value is increased by 2, what would be the effect on the corresponding y-value?
A. The value of y will be 3 times as large.
B. The value of y will decrease to be ½ as large.
C. The value of y will increase by 6.
D. The value of y will decrease by 6.
4.02 Graph, factor, and evaluate quadratic functions to solve problems.
Skill
Always look for a GCF first!!!!!
A. Factoring
1. Find two factors of c that have a sum of b.
2. (x + ____)(x + _____)
B. Factoring
1. Multiply a × c.
2. Rewrite bx as 2 terms.
3. Solve by grouping.
C. Graphing – Look for the x-intercepts of the graph.
Apply
The function describes newspapers circulation (millions) in the United States for 1920-98 (x = 20 for 1920).
Use the graphing calculator.
A. Examine the table to see between 20 and 98 where the values of y are increasing or decreasing.
B. What is the maximum value of the graph?
C. In the table, look for the value of x when y is around 45.
Practice
1. Look for all possible y-values for the function. The graph will have a maximum value since a is a negative number (a = -3).
2. Substitute 240 for v0. Look at the highest x-intercept.
3. The number of real roots corresponds to the number of x-intercepts.
Model the Process
1. Factor
(x + ______) (x – ______)
2. Factor
2x2 – ______+ ______– 9
______( _____ – _____) + _____ (_____ – ______)
(2x – _____)(x + ______)
Apply
A. Identify periods of increasing circulation and decreasing circulation.
B. According to the function, when did newspaper circulation peak?
C. When will circulation approximate 45 million?
Practice
1. Given, what is the range of the function?
A. all real numbers less than or equal to 5
B. all integers less than or equal to 5
C. all nonnegative real numbers
D. all nonnegative integers
2. An object is fired upward at an initial velocity, v0, of 240 ft/s. The height, h(t), of the object is a function of time, t, in seconds and is given by the formula . How long will it take the object to hit the ground after takeoff?
A. 16 seconds B. 15 seconds
C. 7.5 seconds D. 4 seconds
3. Which equation has two real roots?
A. B.
C. D.
4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.
Skill
Solving systems of equations and inequalities by graphing
1. Graph both equations/inequalities
2. The intersection point is the solution or the common shaded region is the solution region.
Solving systems using algebra
A. Substitution
1. One equation is or can be solved for one variable.
2. Substitute the value into the other equation.
3. Solve for the remaining variable.
4. Solve for the other variable.
B. Elimination
1. Both equations are in standard form.
2. Coefficients of one of the variables are the same or opposite.
3. Add/Subtract equations together to eliminate a variable.
4. Solve for the remaining variable.
5. Solve for the other variable.
Apply
During the band’s fruit sale, five dozen oranges cost as much as four dozen grapefruits. Terry bought two dozen oranges and a dozen grapefruit spending $27.30.
A. Define your variables.
B. Write an equation that relates the number of oranges to the number of grapefruits.
C. Write an equation that shows how much was spent on fruit.
D. Solve the system.
Practice
1. Use the application as guidance. Write two equations. One for the sales and one relating cost.
2. Use the calculator.
Model the process
Solve exactly without a calculator.
Use elimination. Multiply the top equation by -3
Solve for x.
______
______y = ______
y = ______
(______, _____)
Apply
A. n → oranges
g → ______
B. _____ n = _____ g
C. _____n + _____ g = ______
D. What was the cost of a dozen oranges?
Practice
1. A store received $823 from the sale of 5 tape recorders and 7 radios. If the receipts from the tape recorders exceeded the receipts from the radios by $137, what is the price of a tape recorder?
A. $49 B. $68
C. $84 D. $96
2. A region is defined by this system:
In which quadrants of the coordinate plane is the region located?
A. I, II, III only B. II, III only
C. III, IV only D. I, II, III, IV
4.04 Graph and evaluate exponential functions to solve problems.
Skill
Substitute using exponential equation formulas.
· Compound Interest
A → New amount after interest
P → Initial amount deposited
r → rate (decimal)
t → time (years)
· Exponential form
a → y-intercept
b → growth (b > 0)
decay (b < 0) ratio
· Geometric Sequence
an→ nth term
a1 → 1st term
n→ number of terms
r → common ratio between consecutive terms
Apply
The average weekly food cost for a family of four in 1990 was $128.30. For the next ten years the weekly food cost increased 2.45% annually. The function represents the cost of food for that period.
A. Substitute ______in for x.
B. Find the difference between the average cost in 2006 and 1990.
Practice
1. Substitute $1,000 for ______, ______for r and ______for t.
2. Substitute half of the original price in for V. Look at the rate at which the car is depreciating or substitute each answer until you get half of the purchase price.
Apply
A. Estimate the weekly food cost for a family of four in 2006.
B. How much has food cost increased since 1990?
Practice
1. When Robert was born, his grandfather invested $1000 for his college education. At an interest rate of 4.5%, compounded annually, approximately how much would Robert have at age 18? (use the formula , where P is the principal, r is the interest rate, and t is the time in years)
A. $1,810
B. $2,200
C. $3,680
D. $18,810
2. A new automobile is purchased for $20,000. If gives the car’s value after x years, about how long will it take for the car to be worth half its purchase price?
A. 3 years
B. 4 years
C. 5 years
D. 6 years
GOAL 3
The learner will collect, organize, and interpret data with matrices and linear models to solve problems.
3.01 Use matrices to interpret data.
3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems.
3.03 Create linear models for sets of data to solve problems.
a) Interpret constants and coefficients in the context of the data.
b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
3.01 Use matrices to interpret data
Skill
· Read and interpret information from a matrix.