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Linear Relations and Functions B-2 Linear Equations ACT WARM-UP • Which of the following is equivalent to (b)(b)(b)(b)(b) + (b)(b)(b)(b)(b)(b)? • A) 11b B) 11b C) b + 11 D) b5 + b6 E) b11 • When multiplying like bases, keep the base and add exponents. The answer is D) b5 + b6. Objectives • Identify linear equations and functions. • Write linear equations in standard form and graph them. Essential Question How do you graph linear equations written in standard form? An equation such as x + y = 4 is called a linear equation. A linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. A linear equation does not contain variables with exponents other than 1. The variables are not under a radical sign and cannot be an exponent. Linear Equation • • • • 5x – 3y = 7 x = 9 (not a function) 6a = – 3b – 15 y = 0.5x Not Linear • 7m + 4n² = – 8 • y = √x + 5 • x + xy = 1 • y= 1 𝑥 A linear function is a function whose ordered pairs satisfy a linear equation. Any linear function can be written in the form f(x) = mx + b, (commonly called slopeintercept form) where x is the independent variable and m (slope) and b (y-intercept) are constants. The graph of a linear function is a straight line made up of the set of all points that satisfy y = f(x). State whether Explain. is a linear function. Answer: This is a linear function because it is in the form State whether Explain. is a linear function. Answer: This is not a linear function because x has an exponent other than 1. State whether Explain. is a linear function. Answer: This is a linear function because it can be written as State whether each function is a linear function. Explain. a. Answer: yes; b. Answer: No; x has an exponent other than 1. c. Answer: No; two variables are multiplied together. Meteorology The linear function 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 37C. What is normal body temperature in degrees Fahrenheit? Original function Substitute. Simplify. Answer: Normal body temperature, in degrees Fahrenheit, is 98.6F. Meteorology The linear function can be used to find the distance m(s) in miles from a storm, based on the number of seconds s that it takes to hear thunder after seeing lightning. a. If you hear thunder 10 seconds after seeing lightning, how far away is the storm? Answer: 2 miles b. If the storm is 3 miles away, how long will it take to hear thunder after seeing lightning? Answer: 15 seconds Standard Form of a Linear Equation • The standard form of a linear equation is Ax + By = C, where • A ≥ 0, • A and B are not both zero at the same time, • A, B, and C are integers, and • A, B, and C do not have a GCF other than 1. Write in standard form. Identify A, B, and C. Original equation Add – 3x to each side. Multiply each side by –1 so that A 0. Answer: and Write and C. in standard form. Identify A, B, Original equation Add – 2y to each side. Multiply each side by –3 so that the coefficients are all integers. Answer: and Write and C. in standard form. Identify A, B, Original equation Add – 4 to each side. Divide each side by 2 so that the coefficients have a GCF of 1. Answer: and Write each equation in standard form. Identify A, B, and C. a. Answer: and b. Answer: and c. Answer: and You should be familiar with graphing a function by making a table of values, seeing a pattern, and connecting the points with a line or smooth curve. Since two points determine a line, there are quicker ways to graph a linear function. One way is to find the points at which the graph intersects each axis and connect them with a line. The ycoordinate of the point at which a graph crosses the y-axis is called the y-intercept (0, b). The x value at this point is zero. Likewise, the xcoordinate of the point at which it crosses the xaxis is the x-intercept (x, 0). The y value at this point is zero. Find the x-intercept and the y-intercept of the graph of Then graph the equation. The x-intercept is the value of x when Original equation Substitute 0 for y. Add 4 to each side. Divide each side by –2. The x-intercept is –2. The graph crosses the x-axis at (–2, 0). Likewise, the y-intercept is the value of y when Original equation Substitute 0 for x. Add 4 to each side. The y-intercept is 4. The graph crosses the y-axis at (0, 4). Use the ordered pairs to graph this equation. Answer: The x-intercept is –2, and the y-intercept is 4. (0, 4) (–2, 0) Find the x-intercept and the y-intercept of the graph of Then graph the equation. Answer: The x-intercept is –2, and the y-intercept is 6. Essential Question How do you graph linear equations written in standard form? Find the x- and y-intercepts by substituting 0 for x and y respectively. Plot the points on the appropriate axis and connect with a straight line extending beyond the points. Math Humor • Parent: Did you study your algebra lesson at the family reunion? • Student: Sure, it was a function with relations.