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## Linear Equations and Functions

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**Linear Equations and Functions**Chapter 2 Mr. Hardy**DO NOW**• Complete the Personal Identity Worksheet- ANONYMOUSLY • Take out the Chapter 1.4 Practice B Assignment**DO NOW**• Skill Review on Page 66 in the textbook!**Journal Topics**• If 1/3 of a number is 2 more than 1/5 of the number, then what is an equation that can be used to find the number x? • If a six-sided polygon has 2 sides of length x – 2y each and 4 sides of length 2x + y, what is its perimeter?**ACT Practice**• A book contains 10 photographs, some in color and some in black-and-white. Each of the following could be the ratio of color to black-and-white photographs EXCEPT: • A) 9:1 • B) 4:1 • C) 5:2 • D) 3:2 • E) 1:1**Chapter 2 Intro**• Topics • Functions and their Graphs • Slope and Rate of Change • Quick Graphs of Linear Equations • Writing Equations of Lines • Correlation and Best-Fitting Lines • Linear Inequalities and Two Variables • Absolute Value Functions**Relations vs Functions**• A relation is a mapping, or pairing, of input values with output values. Simply, it is any ordered pair that shows a relationship between an input and output value • Domain – input values • Range – output values • (D, R) • A relation is a FUNCTION provided there is exactly one output for each input (one x-value yields one and only one y-value)**Domain and Range**• Find the Domain and Range of the ordered pairs • (0, 9.1) • (10, 6.7) • (20, 10.7) • (30, 13.2) • (34, 15.5)**Functions in Context**• What is the domain and range? • Based on what we know, could this chart represent a function?**Take a Better Look**This IS NOT a Function This IS a Function**More Examples**This IS a Function This IS a Function**Observe the Following Number Sets**THIS IS A FUNCTION THIS IS NOT A FUNCTION**Relations Using Ordered Pairs**• Ordered Pairs come in the form of (x, y) • Ordered Pairs can be plotted as points on a coordinate plane • Describe the x and y values in each quadrant**Vertical Line Test**• The vertical line test is a trick used to determine whether or not a graph represents a function • Simply draw a vertical line down the graph. If it goes through more than one (two or more) points on the graph, it does NOT represent a function**Vertical Line Test**BY THE VERTICAL LINE TEST, THIS IS NOT A FUNCTION**Graphing a Function**• Graph the Function y = ½ x + 1 (linear function) or in function notation f(x) = ½x + 1 • We can begin by constructing a table • Plot the Points • Draw a line**Example Evaluating f(x)**• F(x)= 2x + 1 • 2 (0) + 1 = 1 (0, 1) • 2 (1) + 1 = 3 (1, 3) • 2 (2) + 1 = 5 (2, 5) • 2 (6) + 1 = 13 (6, 13) • Every input will result in one output; therefore, this is a function**Evaluating F(x)**• F(x) = √x • √1 = ±1 • √4 = ±2 • √9 = ±3 • √100 = ±10 • √144 = ±12 • Every input will result in two outputs (± the squares); consequently, this is NOT a function!!!!!!!**Linear Functions**• y = mx + b • Function notation • f(x) = mx + b • The symbol f(x) is read as “f of x,” meaning the value of the function at x. • This is another name for y! • Think in terms of independent and dependent variables.**GO TO THE BOARD**• Draw or write an example of a function and an example that is NOT a function. • A table • A map • Coordinates • Or Graph**Wrapping Up**• Is a function always a relation? Is a relation always a function? Explain your reasoning • Homework: Chapters 2.1 Numbers 20 - 42 even, 49 and 50, and 53 - 58 all. • Read chapter 2.2 and take notes!!!**DO NOW: Homework Quiz**• Chapter 1 Review (pg 58 – 60) • 6, 15, 18, 22, 32 • Section 2.1 (71 – 73) • 30 • ANSWERS/GRAPHS ONLY**Recall**• Solve the equation • -⅓(x – 15) = -48 • 6x + 5 = 0.5(x + 6) – 4 • Simplify, when x = 6, and y = 4**Evaluate the Function**• Evaluate f(x) = -3x2 – 2x + 8 • when x = -2 • Evaluate f(5) = -½x - 9**Introduction**• Complete the Chart and Analyze a trend or pattern based on gas prices. Graph the ordered pairs if necessary**WHAT DID YOU NOTICE?**• Graph the ordered pairs- Label the axes. • What does the horizontal axis represent? • What does the vertical axis represent?**WHAT ELSE DID YOU NOTICE?**• What is the pattern that you noticed? • What is the ratio of the vertical change to the horizontal change? Simplified. • What does this represent? • Does it remain constant based on the gas prices? • Check other ratios to find out.**Slope**A line has a positive slope if it is going uphill (increasing) from left to right. A line has a negative slope if it is going downhill (decreasing) from left to right.**Slope**• The steepness of a line that compares the RATE of CHANGE (The change in y per unit x). The larger the slope the steeper the line • Random information: • Slope is usually denoted by • The m comes from the French verb monter, meaning “to rise” or “to ascend.”**Classification of Slope**• A line with a positive slope rises (increases) from left to right (m > 0). • A line with a negative slope falls (decreases) from left to right (m < 0). • A line with a slope of zero is horizontal (m = 0) or y = b • A line with an undefined slope is vertical (m is undefined) or x = a**Parallel and Perpendicular Lines**• Parallel lines do not intersect, and have the same slope • Perpendicular lines intersect, and form a right angle. The lines are perpendicular if AND ONLY if their slopes are negative reciprocals of one another • Ex. m = ½ and m = -2 • Give another example! • Try guided practice 12 – 15**What’s the Slope?**When given the graph, it is easier to apply “rise over run”.**Determine the slope of the line.**Start with the lower point and count how much you rise and run to get to the other point! rise 3 = = run 6 6 3 Notice the slope is positive AND the line increases!**Slope and Rate of Change**• In 2008, 23% of students at RHS were not proficient in mathematics. In 2012, 14% were not proficient. Find the rate of change.**Savings Account**• Michael started a savings account with $300. After 4 weeks, he had $350 dollars, and after 8 weeks, he had $400. What is the rate of change of money in his savings account per week? • Find out how much money he would have after a year and a half.**Find the Slope**• (-1, 4) and (1, -2) • (-5, 3) and (-6, -1) • (0, 5) and (4, ½)**Pop Quiz**• Draw a map that represents function and a graph that represents a non-function. SHOW AND EXPLAIN WHY they represent functions/non-functions**Homework**• Chapter 2.2 • 18 - 30 all, 38-46 all, 55-58 all • Read 2.3 and take notes!**Chapter 2.3**Mr. Hardy Algebra 2**DO NOW**• Solve the equation • |x – 10| =17 • Solve for h • S = 2πrh + 2πr2 • Find the slope of a line given points: • (-⅕, ⅝) and (¾, -⅔)**Recall: Slope as a Rate of Change**• You are driving from Grand Rapids, MI to Detroit, MI. You leave Grand Rapids at 4:00pm. At 5:10 pm you pass through Lansing, MI, a distance of 65 miles. • Approximately what time will you arrive in Detroit if it is 150 miles from Grand Rapids?**Discussion: Illegal Drug Use**• Graph shows illegal drug use by age group. Find the slope of the line segment for ages 12-17. Describe what is means in practical terms.**Chapter 2.3- Quick Graphs**• y-intercept • (0, b) • To find the y-intercept of a line, let x = 0 in the equation, then solve for y. • x-intercept • (a, 0) • To find the x-intercept of a line, let y = 0 in the equation, then solve for x.