Algebra I Notes Slope Unit 04a
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1 OBJECTIVE: F.IF.B.6 Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F.LE.A. Construct and compare liner, quadratic, and exponential models and solve problems. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. BIG IDEA: The concept of slope is important because it is used to measure the rate at which changes are taking place. In real-life problems, we often need to explore and understand how things change and about how one item changes in response to a change in another item. PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY: rate of change: a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable rise: the difference in the y-values of two points on a line run: the difference in the x-values of two points on a line slope: the ratio of rise to run for any two points on a line SKILLS: find rates of change and slopes relate a constant rate of change to the slope of a line REVIEW AND EXAMPLES: The slope of a line is the constant rate of change occurring as you move along the line from left to right (the steepness of the line). There are four types of slope: positive negative zero undefined as x increases, y increases as x increases, y decreases as x increases, y is constant x is constant Alg Unit 04a Notes Slope page of 0 6/0/3
2 Finding slope given a graph: Ex. Find the slope of the given line. Step One: find two points that fall on the line. Step Two: find the ratio of vertical change to horizontal change (from left to right) between the two points. m = rise run = verticalchange horizontal change m = rise run = Find the slope of a line using Slope Formula. If you don t have a graph of the line, you can find the slope of the line using the slope formula when given two points; ( x, y) and ( x2, y 2) : change in y m change in x y x 2 y x 2 Ex 2. Find the slope of a line that goes through the points (3, 4) and (5, 7). Substitute in the given values in the ordered pairs into the slope formula and simplify. m change in y 7 (4) 3 3 change in x 5 (3) 2 2 Note: Do not express slope as a mixed number. Leave it in simplified fraction form. Alg Unit 04a Notes Slope page 2 of 0 6/0/3
3 *How is the slope like unit rate? Any slope is a ratio comparing the change in y to the change in x. A rate is a unit rate if it has a denominator of. In the examples above, the slope is 3. Which tells us that as y increases by 2 three units, x increases by two units. But, another way to look at is that as y increases by 3 2 units, x increases by one unit. Let s look at the line in the first example. As x increases by unit (from 3 to 4), y increases by.5 units (from 4 to 5.5) Let s look at a table of values for some points that fall on the graph of that linear function. Notice that each time x increases by unit, y increases by.5 units. x y The slope is the unit rate of the function. It shows the rate of change vertically, as the graph moves one unit to the right. Alg Unit 04a Notes Slope page 3 of 0 6/0/3
4 Ex 3. Use the table of values to find the rate of change and then explain its meaning. number of video games x total cost ($) y Step One: select any two points to use in the slope formula. (2, 78) and (4, 56) 56 (78) 78 Step Two: calculate the slope m 39 4 (2) 2 Step Three: explain what in means in the context of the problem. It means that $39 is the cost per game or the unit rate. Ex 4. Determine if the function is linear. Explain your answer. x y Step One: Step Two: Step Three: Pick two points and find the slope between them. ( 3, 0) and (, 2) 2 (0) 2 m ( 3) 2 Pick two other points and find the slope between them. (, 2) and (, 6) 6 (2) 4 m 2 ( ) 2 Determine if it s linear. It is not linear because the slope is not constant. Alg Unit 04a Notes Slope page 4 of 0 6/0/3
5 Why does the slope have to be constant in a linear equation? Because, if the slope is not constant than it is not a linear function. Let s look at that on a graph. Notice that if we start at the point ( 7, 5) and follow a slope of, it takes us to the point ( 4, 4). If we follow the same slope 3 again, we get to the point (, 3) and then to the point (2, 2). All of these points fall on a straight line. But, if we change the slope (pattern) to 4, we do not get to a point that falls on the same line as all of the other points. Comparing slopes: Ex 5. Look at the graph below. Using the line with a slope of 2 as the original line, describe how the steepness of the line changes as the slope changes. 4 m m 2 m = 0 2 *The line is much steeper than the original line. It is rising 4 times quicker while running the same amount. *The line is less steep than the original line. It is rising half as quickly, and running twice as far. *The line is flat. It is not rising at all as it runs to the right. *The line has the same steepness as the original line, except it is going downward. Alg Unit 04a Notes Slope page 5 of 0 6/0/3
6 Relating Slope to Similar Triangles: Similar triangles are proportional. *All corresponding ratios are proportional. 4 is to 8 as 2 is to Let s put the similar triangles on a coordinate plane. Notice that both triangles hypotenuses fall on the same straight line. The legs of each triangle are like the rise and the run of the slope of the line. The purple triangle s rise is 4 and its run is 8. The red triangle s rise is 2 and its run is 4. Which means that both slopes reduce to an equivalent slope of 2. *Using similar triangles we can prove that no matter what two points we choose on a line, we will always find the same slope between those points. Ex 6. Find the value of y so that the line passing through the points ( 2,) and (4, y) has a slope of 2 3. y2 y Step One: write the slope formula; m x x 2 Step Two: plug in the given values into the formula; 2 y () 3 4 ( 2) 2 y Step Three: solve for y; = y y = y 3 6 Alg Unit 04a Notes Slope page 6 of 0 6/0/3
7 ASSESSMENT ITEMS:. Find the slope of the line. ANS: rise 3 run 3 2. Find the slope of the line. ANS: rise run What is the slope of a line that goes through the points ( 2, 5) and (6, )? A. 2 B. 2 C. 2 D. 2 ANS: B 4. What is the slope of a line that goes through the points (5, 0) and (7, 8)? A. 4 B. 4 C. 4 D. 4 ANS: D Alg Unit 04a Notes Slope page 7 of 0 6/0/3
8 5. Find the slope of the line. ANS: rise 0 0 run 3 6. Find the slope of the line. ANS: rise 4 undefined run 0 7. What is the slope of a line that goes through the points (6, 3) and (6, )? A. 3 B. 3 C. 0 D. undefined ANS: D 8. What is the slope of a line that goes through the points (, 5) and (4, 5)? A. 3 B. 3 C. 0 D. undefined ANS: C Alg Unit 04a Notes Slope page 8 of 0 6/0/3
9 9. Given the table of values below, determine whether or not the points all fall on a straight line. Explain your answer. x y ANS: Yes, it is linear. The slope is constant from point to point and equals Find the value of y so that the line passing through the points (0,3) and (4,y) have a slope of 3. A. 5 B. 9 C. 9 D. 5 ANS: D. Use the table of values to find the rate of change and then explain its meaning. driving time (h) distance traveled (m) ANS: 38/, which means the car is traveling at a speed of 38 mph. Alg Unit 04a Notes Slope page 9 of 0 6/0/3
10 2. Use the table of values to find the rate of change and then explain its meaning. number of floor tiles x Area of tiled surface (in 2 ) y ANS: 6/, which means that each floor tile is 6 in When driving down a certain hill, you descend 5 feet for every 000 feet you drive forward. What is the slope of the road? ANS: The point (, 8) is on a line that has a slope of 3. Is the point (4, 7) on the same line? Explain your reasoning. ANS: Yes, the slope between the two points is also In 996, a company had a profit of $53,000,000. In 2002, the profit was $86,000,000. If the profit increased the same amount each year, find the average rate of change of the company s profit in dollars per year. ANS: $5,500,000 per year Alg Unit 04a Notes Slope page 0 of 0 6/0/3
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