Midterm Review

Reviewing Linear Equations

Finding the Slope Given a graph how do I find the slope? Find the Slope of the following graphs

Finding the Slope Find the Slope of the following graphs What about the slope of these graphs?

Finding the Slope What is the formula for finding the slope? Find the slope of the line through the given points. (-1, 2) and (-5,10) 2. (-7, 10 ) and (1, 10) 3. (1,3) and (0,-9) 4. (3, 7) and (3, -8)

Direct Variation

Direct Variation Direct Variation: the relationship that can be represented by a function if the form: y = kx Constant of variation: the constant variable K is the coefficient of x on the y=kx equation.

Identifying a Direct Variation: If the equation can be written in y = kx we have a direct variation. Ex: Does the equation represent a direct variation? a) 7y = 2x b) 3y + 4x = 8

Writing a direct variation equation Suppose y varies directly with x, and y = 35 when x = 5. Write a direct variation equation. Then find y when x = 9.

Writing a direct variation equation Suppose y varies directly with x, and y = 10 when x = -2. Write a direct variation equation. Then find y when x = -15.

Writing a direct variation from a table For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.

Writing a direct variation from a table For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.

Graphing Linear Equations Graph the following linear equation. Y = 𝟏 𝟐 𝒙 + 3 What form of linear equation should be used to graph? What are our steps to graph a linear equation?

Graphing Linear Equations Graph the following linear equation. Y = -2𝒙 + 1

Changing Forms For practice, change the following equations to the slope-intercept form:   6x + y = 5 6 + 2y = 8x 2y = 6x – 2 4x – 2y = 12 What is the standard form of a linear equation? What are the special restrictions that go along with the standard form?

Changing Forms What is the point-slope form of a linear equation? Change the following to standard form. y – 5 = -1/2 (x – 3) y – 3 = ½(x + 6)

Writing Equations Write the equation given a point and the slope in point-slope form. Point: (-7, 2) Slope: 3

Writing Equations Write the equation of a line given two points. (-8, 3) (-4, 4)

Parallel & Perpendicular Lines

Parallel Lines Parallel Lines - lines that never intersect What is the slope of the red line? 1/2 blue line? Parallel Lines have the slope! same

Parallel Lines Determine if the lines are parallel yes

Equations of Parallel Lines Write an equation for the line that contains (5, 1) and is parallel to m =

Equations of Parallel Lines Write an equation for the line that contains (2, -6) and is parallel to m = 3 y + 6 = 3(x – 2)

Perpendicular Lines Perpendicular Lines – lines that intersect to form right angles What is the slope of the red line? -1/4 blue line? 4/1 Perpendicular Lines have slope! opposite reciprocal

Perpendicular Lines What is the slope of the perpendicular line? 7. -5/2 8. 5/1 = 5 9. 1/2

Equations of Perpendicular Lines Find the equation of the line that contains (0, -2) and is perpendicular to y = 5x + 3 m =

Equations of Perpendicular Lines Find the equation of the line that contains (2, -3) and is perpendicular to m = 2 y + 3 = 2(x – 2)

Challenge What is the slope of the line that is parallel to x = 4? undefined What is the slope of the line that is perpendicular to x = 4? zero

Arithmetic Sequences

An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

The common difference is the amount it increase or decreases by each time.

Formula for the nth term of an ARITHMETIC sequence.

Arithmetic sequences The sequence below shows the total number of days Francisco had used his gym membership at the end of weeks 1, 2, 3, and 4. Assuming the pattern below continued, which function (equation) could be used to find the total number of days Francisco had used his gym membership at the end of week n? 5, 10, 15, 20…

Arithmetic sequences Consider the following arithmetic sequence: 6, 11, 16, 21… What is the value of the 23rd term?

Determine the explicit formula for the following

Scatterplots & Trend Lines

DETERMINING THE CORRELATION OF X AND Y TYPES OF CORRELATION Positive Correlation Negative Correlation No Correlation

Scatter plots provide a convenient way to determine whether a ___________ exists between two variables. correlation positive A __________ correlation occurs when both variables increase. negative A ___________ correlation occurs when one variable increases and the other variable decreases. If the data points are randomly scattered there is _______ or no correlation. little

Determine the Correlation Write increase or decrease to complete the sentence regarding expectations for the situation. The more one studies, their grades will _____. The more a person diets, their weight will ______, The longer one diets, the amount of weight loss will ______. The more a person exercises, their muscle mass will _____. The more a person exercises, their fat mass will ______. The more a person jumps rope, their heart rate will _____. The more a class talks, the teacher’s patience will _____. The more a person reads, their vocabulary will _____. This is called correlation.

Line of Best Fit line of best fit - The trend line that shows the relationship between two sets of data most A graphing calculator computes the equation of a line of best fit using a method called linear regression. The graphing calculator gives you the correlation coefficient r. -1 0 1 negative correlation no correlation positive correlation

Finding the Equation of Line of Best Fit Example 1: Find an equation for the trend line and the correlation coefficient. Estimate the number of calories in a fast- food that has 14g of fat. Show a scatter plot for the given data. Calories and Fat in Selected Fast-Food Meals Fat(g) 6 7 10 19 20 27 36 Calories 276 260 220 388 430 550 633 Solution: Use your graphing calculator to find the line of best fit and the correlation coefficient.

Cont (example 1)... Step 2. Use the CALC feature in the STAT screen. Find the equation for the line of best fit  LinReg (ax + b) LinReg y = ax + b a = 13.60730858 b = 150.8694896 r2 = .9438481593 r = .9715184812 Slope y-intercept Correlation coefficient The equation for the line of best fit is y = 13.61x + 150.87 and the correlation coefficient r is 0.9715184812

Finding the Equation of Line of Best Fit Recreation Expenditures Example 2. Use a graphing calculator to find the equation of the line of best fit for the data at the right. What is the correlation coefficient? Estimate the recreation expenditures in 2010. Year Dollars (Billions) 1993 340 1994 369 1995 402 1996 430 1997 457 1998 489 1999 527 2000 574 Answers: y = 32.33x - 2671.67 r = 0.9964509708 The expenditures in 2010 will be 885 billions Let 1993 = 93

Do these… Use a graphing calculator to find the equation of line of best fit for the data. Find the value of the correlation coefficient r. 3. Use the equation to predict the time needed to travel 32 miles on a bicycle. How many miles will he travel for 125 mins. Speed on a Bicycle Trip Miles 5 10 14 18 22 Time (min) 27 46 71 78 107 Answers: y = 4.56x + 2.83 r. = 0.9881161783 about 148.85 min or 149 min 26.77 or 27 miles.

Systems of Equations

Solve by graphing 𝑦= 2 3 𝑥+ 2 3 𝑦=−4𝑥+24

Solve by substitution 2𝑥−3𝑦=−2 𝑦=−4𝑥+24

Solve by elimination 4𝑥−3𝑦=25 −3𝑥+8𝑦=10

Systems of Inequalities

Write the following system of inequalities Answer:

Write the following system of inequalities Answer:

Exponent Rules

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Let’s Try Some More Complicated Problems

Find the Area

Find the area of the triangle?

Polynomials

Polynomials Write each polynomial in standard form. Then name each polynomial based on degree and number of terms. 1. 4x + x2  1 2. w + 3  2w + 8w3

Adding Polynomials You can add polynomials by adding like terms Ex. (x2 + 3x  2) + (4x2  5x + 2) Identify the like terms There are two Methods. You can add Vertically or Horizontally Let’s try both ways

Subtracting Polynomials Recall that subtraction means to add the opposite. So when you subtract a polynomial, change each of the terms to its opposite. Then add the coefficients. Let’s Try an example Ex. (8t2 + t + 10)  (9t2  9t  1)

More Practice!!!! (5x3 + 3x2  7x + 10)  (3x3  x2 + 4x  1) (4m3 + 7m  4) + (2m3  6m + 8) (7c3 + c2  8c  11)  (3c3 + 2c2 + c  4)

More Practice Find the Perimeter 𝒙 𝟐 +𝟐 6x - 3 x + 12 8x-1 𝒙 𝟐 −𝒙

Multiplying Polynomials Let’s try multiplying a monomial and a trinomial! Practice 1: –3c (8 + 2c – c 3) Practice 2: 8a 2 (–a 7 + 7a – 7)

Multiplying Binomials Ex. (2x +1)(x + 2) Box Method F.O.I.L Method

Practice Simplify each product (Use any method) 1. (x + 3)(x + 8) 3. (5h – 3)(h + 7)

More Practice Find the Area 𝒙 𝟐 +𝟐 6x - 3 8x-1

Factoring

Factor out the GCF

Factor out the GCF

Factoring the Difference of Squares

Factoring the Difference of Squares

Factoring when a = 1

Factoring when a = 1 *Don’t forget to check for a GCF

Factoring when a>1

Factoring Perfect Square Trinomials

Factoring Perfect Square Trinomials

Factoring by Grouping Ex:

Factoring by Grouping Ex2:

You Try!

Answers