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Five-Minute Check (over Lesson 2–1) CCSS Then/Now New Vocabulary Example 1: Identify Linear Functions Example 2: Real-World Example: Evaluate a Linear Function Key Concept: Standard Form of a Linear Equation Example 3: Standard Form Example 4: Use Intercepts to Graph a Line Lesson Menu

A. function; one-to-one B. function; onto Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 1

A. function; one-to-one B. function; onto Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 1

A. function; one-to-one B. function; onto C. function; both Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 2

A. function; one-to-one B. function; onto C. function; both Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 2

A. function; one-to-one B. function; onto Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. {(1, 2), (2, 1), (5, 2), (2, 5)}. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 3

A. function; one-to-one B. function; onto Determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. {(1, 2), (2, 1), (5, 2), (2, 5)}. A. function; one-to-one B. function; onto C. function; both D. not a function 5-Minute Check 3

Find f(–3) if f(x) = x2 + 3x + 2. A. 20 B. 10 C. 2 D. –2 5-Minute Check 4

Find f(–3) if f(x) = x2 + 3x + 2. A. 20 B. 10 C. 2 D. –2 5-Minute Check 4

What is the value of f(3a) if f(x) = x2 – 2x + 3? A. 3a + 3 B. 3a2 – 6a + 3 C. 9a2 – 2a + 3 D. 9a2 – 6a + 3 5-Minute Check 5

What is the value of f(3a) if f(x) = x2 – 2x + 3? A. 3a + 3 B. 3a2 – 6a + 3 C. 9a2 – 2a + 3 D. 9a2 – 6a + 3 5-Minute Check 5

Mathematical Practices Content Standards F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS

You analyzed relations and functions. Identify linear relations and functions. Write linear equations in standard form. Then/Now

linear relation nonlinear relation linear equation linear function standard form y-intercept x-intercept Vocabulary

Identify Linear Functions A. State whether g(x) = 2x – 5 is a linear function. Write yes or no. Explain. Answer: Example 1A

Identify Linear Functions A. State whether g(x) = 2x – 5 is a linear function. Write yes or no. Explain. Answer: Yes; this is a linear function because it is in the form g(x) = mx + b; m = 2, b = –5. Example 1A

Identify Linear Functions B. State whether p(x) = x3 + 2 is a linear function. Write yes or no. Explain. Answer: Example 1B

Identify Linear Functions B. State whether p(x) = x3 + 2 is a linear function. Write yes or no. Explain. Answer: No; this is not a linear function because x has an exponent other than 1. Example 1B

Identify Linear Functions C. State whether t(x) = 4 + 7x is a linear function. Write yes or no. Explain. Answer: Example 1C

Identify Linear Functions C. State whether t(x) = 4 + 7x is a linear function. Write yes or no. Explain. Answer: Yes; this is a linear function because it can be written as t(x) = mx + b; m = 7, b = 4. Example 1C

A. State whether h(x) = 3x – 2 is a linear function. Explain. A. yes; m = –2, b = 3 B. yes; m = 3, b = –2 C. No; x has an exponent other than 1. D. No; there is no slope. Example 1A

A. State whether h(x) = 3x – 2 is a linear function. Explain. A. yes; m = –2, b = 3 B. yes; m = 3, b = –2 C. No; x has an exponent other than 1. D. No; there is no slope. Example 1A

B. State whether f(x) = x2 – 4 is a linear function. Explain. A. yes; m = 1, b = –4 B. yes; m = –4, b = 1 C. No; two variables are multiplied together. D. No; x has an exponent other than 1. Example 1B

B. State whether f(x) = x2 – 4 is a linear function. Explain. A. yes; m = 1, b = –4 B. yes; m = –4, b = 1 C. No; two variables are multiplied together. D. No; x has an exponent other than 1. Example 1B

C. State whether g(x, y) = 3xy is a linear function. Explain. A. yes; m = 3, b = 1 B. yes; m = 3, b = 0 C. No; two variables are multiplied together. D. No; x has an exponent other than 1. Example 1C

C. State whether g(x, y) = 3xy is a linear function. Explain. A. yes; m = 3, b = 1 B. yes; m = 3, b = 0 C. No; two variables are multiplied together. D. No; x has an exponent other than 1. Example 1C

f(C) = 1.8C + 32 Original function f(37) = 1.8(37) + 32 Substitute. Evaluate a Linear Function A. METEOROLOGY The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit f(C) that are equivalent to a given number of degrees Celsius C. On the Celsius scale, normal body temperature is 37C. What is it in degrees Fahrenheit? f(C) = 1.8C + 32 Original function f(37) = 1.8(37) + 32 Substitute. = 98.6 Simplify. Answer: Example 2

f(C) = 1.8C + 32 Original function f(37) = 1.8(37) + 32 Substitute. Evaluate a Linear Function A. METEOROLOGY The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit f(C) that are equivalent to a given number of degrees Celsius C. On the Celsius scale, normal body temperature is 37C. What is it in degrees Fahrenheit? f(C) = 1.8C + 32 Original function f(37) = 1.8(37) + 32 Substitute. = 98.6 Simplify. Answer: Normal body temperature, in degrees Fahrenheit, is 98.6°F. Example 2

Divide 180 Fahrenheit degrees by 100 Celsius degrees. Evaluate a Linear Function B. METEOROLOGY The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit f(C) that are equivalent to a given number of degrees Celsius C. There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree? Divide 180 Fahrenheit degrees by 100 Celsius degrees. Answer: Example 2

Divide 180 Fahrenheit degrees by 100 Celsius degrees. Evaluate a Linear Function B. METEOROLOGY The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit f(C) that are equivalent to a given number of degrees Celsius C. There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree? Divide 180 Fahrenheit degrees by 100 Celsius degrees. Answer: 1.8°F = 1°C Example 2

A. 50 miles B. 5 miles C. 2 miles D. 0.5 miles Example 2A

A. 50 miles B. 5 miles C. 2 miles D. 0.5 miles Example 2A

A. 0.6 second B. 1.67 seconds C. 5 seconds D. 15 seconds Example 2B

A. 0.6 second B. 1.67 seconds C. 5 seconds D. 15 seconds Example 2B

Concept

Write y = 3x – 9 in standard form. Identify A, B, and C. y = 3x – 9 Original equation –3x + y = –9 Subtract 3x from each side. 3x – y = 9 Multiply each side by –1 so that A ≥ 0. Answer: Example 3

Write y = 3x – 9 in standard form. Identify A, B, and C. y = 3x – 9 Original equation –3x + y = –9 Subtract 3x from each side. 3x – y = 9 Multiply each side by –1 so that A ≥ 0. Answer: 3x – y = 9; A = 3, B = –1, and C = 9 Example 3

Write y = –2x + 5 in standard form. A. y = –2x + 5 B. –5 = –2x + y C. 2x + y = 5 D. –2x – 5 = –y Example 3

Write y = –2x + 5 in standard form. A. y = –2x + 5 B. –5 = –2x + y C. 2x + y = 5 D. –2x – 5 = –y Example 3

The x-intercept is the value of x when y = 0. Use Intercepts to Graph a Line Find the x-intercept and the y-intercept of the graph of –2x + y – 4 = 0. Then graph the equation. The x-intercept is the value of x when y = 0. –2x + y – 4 = 0 Original equation –2x + 0 – 4 = 0 Substitute 0 for y. –2x = 4 Add 4 to each side. x = –2 Divide each side by –2. The x-intercept is –2. The graph crosses the x-axis at (–2, 0). Example 4

Likewise, the y-intercept is the value of y when x = 0. Use Intercepts to Graph a Line Likewise, the y-intercept is the value of y when x = 0. –2x + y – 4 = 0 Original equation –2(0) + y – 4 = 0 Substitute 0 for x. y = 4 Add 4 to each side. The y-intercept is 4. The graph crosses the y-axis at (0, 4). Example 4

Use the ordered pairs to graph this equation. Use Intercepts to Graph a Line Use the ordered pairs to graph this equation. Answer: Example 4

Use the ordered pairs to graph this equation. Use Intercepts to Graph a Line Use the ordered pairs to graph this equation. Answer: The x-intercept is –2, and the y-intercept is 4. Example 4

A. x-intercept = –2 y-intercept = 6 What are the x-intercept and the y-intercept of the graph of 3x – y + 6 = 0? A. x-intercept = –2 y-intercept = 6 B. x-intercept = 6 y-intercept = –2 C. x-intercept = 2 y-intercept = –6 D. x-intercept = –6 y-intercept = 2 Example 4

A. x-intercept = –2 y-intercept = 6 What are the x-intercept and the y-intercept of the graph of 3x – y + 6 = 0? A. x-intercept = –2 y-intercept = 6 B. x-intercept = 6 y-intercept = –2 C. x-intercept = 2 y-intercept = –6 D. x-intercept = –6 y-intercept = 2 Example 4

End of the Lesson