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Graphing y = nx2 Lesson 5.4.1

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**Graphing y = nx2 5.4.1 California Standards: What it means for you:**

Lesson 5.4.1 Graphing y = nx2 California Standards: Algebra and Functions 3.1 Graph functions of the form y = nx2 and y = nx3 and use in solving problems. Mathematical Reasoning 2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. Mathematical Reasoning 2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. What it means for you: You’ll learn how to plot graphs of equations with squared variables in them. Key words: parabola vertex

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Lesson 5.4.1 Graphing y = nx2 Think about the monomial x2. You can put any number in place of x and work out the result — different values of x give different results. The results you get form a pattern. And the best way to see the pattern is on a graph.

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**Graphing y = nx2 5.4.1 The Graph of y = x2 is a Parabola**

Lesson 5.4.1 Graphing y = nx2 The Graph of y = x2 is a Parabola You can find out what the graph of y = x2 looks like by plotting points. Examples 1 and 2 show you how to plot a graph of y = x2 for positive and negative values of x.

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Lesson 5.4.1 Graphing y = nx2 Example 1 Plot the graph of y = x2 for values of x between 0 and 6. Solution The best thing to do first is to make a table for the integer values of x like the one below. y 3 2 1 x 4 5 6 y (= x2) 9 16 25 36 2 4 6 1 3 5 10 20 30 40 Then you can plot points on a set of axes using the x- and y-values as coordinates, and join the points with a smooth curve. The curve passes through all the values that fit the equation between the integer points. x Solution follows…

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Lesson 5.4.1 Graphing y = nx2 Example 2 Plot the graph of y = x2 for values of x between –6 and 6. Solution The table of values looks like this: –3 –2 –1 x –4 –5 –6 y (= x2) 9 4 1 16 25 36 y 2 4 6 10 20 30 40 –2 –4 –6 And the curve looks like this. This kind of curve is called a parabola. x Solution follows…

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**Graphing y = nx2 5.4.1 Guided Practice**

Lesson 5.4.1 Graphing y = nx2 Guided Practice 1. Which of the following points are on the graph of y = x2? (1, 1), (–1, 1), (–2, –4), (2, 4), (3, 9), (–4, –16), (5, 25), (6, –36) In Exercises 2–5, calculate the y-coordinate of the point on the graph of y = x2 whose x-coordinate is shown. –10 4. – (1, 1), (–1, 1), (2, 4), (3, 9), and (5, 25) 36 100 1 3 1 9 6.25 Solution follows…

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**Graphing y = nx2 5.4.1 Guided Practice**

Lesson 5.4.1 Graphing y = nx2 Guided Practice In Exercises 6–9, calculate the two possible x-coordinates of the points on the graph of y = x2 whose y-coordinate is shown. 4 and –4 5 and –5 7 and –7 and – Solution follows…

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**Graphing y = nx2 5.4.1 You Can Use a Graph to Solve an Equation**

Lesson 5.4.1 Graphing y = nx2 You Can Use a Graph to Solve an Equation Graphs can be useful if you need to solve an equation. Using them means you don’t have to do any tricky calculations — and they often show you how many solutions the equation has. The downside is that it can be impossible to get an exact answer by reading off a graph.

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Lesson 5.4.1 Graphing y = nx2 Example 3 Using the graph of y = x2 in Example 2, solve x2 = 12. Solution Since the graph shows y = x2, you need to find where y = Then you can find the corresponding value (or values) of x. y 2 4 6 10 20 30 40 –2 –4 –6 There are two different values of x that correspond to y = 12, at approximately x = 3.5 and x = –3.5. y = 12 x x = –3.5 x = 3.5 Solution continues… Solution follows…

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Lesson 5.4.1 Graphing y = nx2 Example 3 Using the graph of y = x2 in Example 2, solve x2 = 12. Solution (continued) There are two different values of x that correspond to y = 12 because 12 has two square roots — a positive one (3.5) and a negative one (–3.5). Or you can look at it another way, and say that the numbers 3.5 and –3.5 can both be squared to give 12 (approximately).

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**Graphing y = nx2 5.4.1 Guided Practice**

Lesson 5.4.1 Graphing y = nx2 Guided Practice Use the graph of y = x2 shown below to solve the equations in Exercises 10–13. 10. x2 = 16 11. x2 = 25 12. x2 = 10 13. x2 = 30 x = 4 or x = –4 y 2 4 6 10 20 30 40 –2 –4 –6 x = 5 or x = –5 x = –5.5 x = 5.5 x = –5 x = 5 x = 3.2 or x = –3.2 (approximately) x = –4 x = 4 x = –3.2 x = 3.2 x x = 5.5 or x = –5.5 (approximately) Solution follows…

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**Graphing y = nx2 5.4.1 The Graph of y = nx2 is Also a Parabola**

Lesson 5.4.1 Graphing y = nx2 The Graph of y = nx2 is Also a Parabola The graph of y = x2 is y = nx2 where n = 1. It has the U shape of a parabola. Other values of n give graphs that look very similar.

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Lesson 5.4.1 Graphing y = nx2 Example 4 Plot the graphs of the following equations for values of x between –5 and 5. a) y = 2x2 b) y = 3x2 c) y = 4x2 d) y = x2 1 2 Solution All these equations are of the form y = nx2, for different values of n (2 then 3 then 4 then ). 1 2 The best place to start is with a table of values, just like before. Solution continues… Solution follows…

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Lesson 5.4.1 Graphing y = nx2 Example 4 Plot the graphs of the following equations for values of x between –5 and 5. a) y = 2x2 b) y = 3x2 c) y = 4x2 d) y = x2 1 2 Solution (continued) x 2x2 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 2 8 18 32 50 3x2 3 12 27 48 75 4x2 4 16 36 64 100 ½ x2 0.5 4.5 12.5 The table on the right shows values for parts a)–d). You then need to plot the y-values in each colored column against the x‑values in the first column. Solution continues…

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**Graphing y = nx2 5.4.1 Solution (continued) Lesson Example 4 y 4 –2 –4**

Increasing values of n Decreasing values of n (n = 4) (n = 3) (n = 2) (n = ½) 4 –2 –4 2 3 1 –1 –3 20 40 60 80 100 5 –5 y = 4x2 Solution (continued) x 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 4x2 4 16 36 64 100 x 2x2 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 2 8 18 32 50 x 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 3x2 3 12 27 48 75 x 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 ½ x2 0.5 2 4.5 8 12.5 y = 3x2 y = 2x2 y = x2 1 2 x

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Lesson 5.4.1 Graphing y = nx2 Notice how all the graphs are “u-shaped” parabolas. And all the graphs have their vertex (the lowest point) at the same place, the origin. 4 –2 –4 2 3 1 –1 –3 20 40 60 80 100 5 –5 y = 4x2 y = 3x2 y = 2x2 y = x2 y x In fact, this is a general rule — if n is positive, the graph of y = nx2 will always be a “u-shaped” parabola with its vertex at the origin.

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Lesson 5.4.1 Graphing y = nx2 Also, the greater the value of n, the steeper the parabola will be. 4 –2 –4 2 3 1 –1 –3 20 40 60 80 100 5 –5 y = 4x2 y = 3x2 y = 2x2 y = x2 y x In Example 4, the graph of y = 4x2 had the steepest parabola, while the graph of y = ½ x2 was the least steep.

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**Graphing y = nx2 5.4.1 Guided Practice**

Lesson 5.4.1 Graphing y = nx2 Guided Practice For Exercises 14–17, draw on the same axes the graph of each of the given equations. 14. y = 5x2 15. y = x2 16. y = 10x2 17. y = x2 1 4 10 y 2 4 10 20 30 40 –2 –4 50 60 70 y = 10x2 y = 5x2 y = x2 1 4 y = x2 1 10 x Solution follows…

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**Graphing y = nx2 5.4.1 Guided Practice**

Lesson 5.4.1 Graphing y = nx2 Guided Practice y In Exercises 18–23, use the graphs from Example 4 to solve the given equations. 18. 2x2 = x2 = 25 20. 4x2 = x2 = 10 22. 3x2 = x2 = 42 2 4 10 20 30 40 –2 –4 50 60 70 y = 4x2 y = 3x2 y = 2x2 y = x2 1 x » 3.2 or x » –3.2 x » 2.9 or x » –2.9 1 2 x » 1.9 or x » –1.9 x » 4.5 or x » –4.5 x » 4.8 or x » –4.8 x » 4.6 or x » –4.6 x Solution follows…

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**Graphing y = nx2 5.4.1 Independent Practice 1 3**

Lesson 5.4.1 Graphing y = nx2 Independent Practice Using a table of values, plot the graphs of the equations in Exercises 1–3 for values of x between –4 and 4. 1. y = 1.5x2 2. y = 5x2 3. y = x2 y 4 –2 –4 2 3 1 –1 –3 20 40 60 80 100 y = 5x2 1 3 y = 1.5x2 y = x2 1 3 x Solution follows…

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**Graphing y = nx2 5.4.1 Independent Practice 2 3**

Lesson 5.4.1 Graphing y = nx2 Independent Practice On the same set of axes as you used for Exercises 1–3, sketch the approximate graphs of the equations in Exercises 4–6. 4. y = 2.5x2 5. y = 6x2 6. y = x2 2 3 y 4 –2 –4 2 3 1 –1 –3 20 40 60 80 100 y = 1.5x2 y = 5x2 y = x2 y = 6x2 y = 2.5x2 y = x2 2 3 x Solution follows…

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**Graphing y = nx2 5.4.1 Independent Practice**

Lesson 5.4.1 Graphing y = nx2 Independent Practice 7. If s is the length of a square’s sides, then a formula for its area, A, is A = s2. Plot a graph of A against s, for values of s up to 10. 10 20 40 60 80 100 A s –10 Solution follows…

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**Graphing y = nx2 5.4.1 Independent Practice 2 3 8 5**

Lesson 5.4.1 Graphing y = nx2 Independent Practice 8. On a graph of y = x2, what is the y-coordinate when x = 103? For Exercises 9–12, find the y-coordinate of the point on the graph of y = x2 for each given value of x. 9. x = 10– x = 10–4 11. x = x = 106 10–2 10–8 2 3 8 5 4 9 64 25 Solution follows…

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**Graphing y = nx2 5.4.1 Independent Practice**

Lesson 5.4.1 Graphing y = nx2 Independent Practice For Exercises 13–15, find the x-coordinates of the point on the y = x2 graph for each given value of y. 13. y = 102 14. y = 10–6 15. y = 28 10 and –10 10–3 and –10–3 24 and –24 Solution follows…

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Lesson 5.4.1 Graphing y = nx2 Round Up In this Lesson you’ve looked at graphs of the form y = nx2, where n is positive. The basic message is that these graphs are all u-shaped. And the greater the value of n, the narrower and steeper the parabola is. Remember that, because in the next Lesson you’re going to look at graphs of the same form where n is negative.

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