Download presentation

Presentation is loading. Please wait.

Published byCamden Baron Modified over 2 years ago

1
1 Lesson Graphing y = nx 2

2
2 Lesson Graphing y = nx 2 California Standards: Algebra and Functions 3.1 Graph functions of the form y = nx 2 and y = nx 3 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

3
3 Graphing y = nx 2 Lesson Think about the monomial x 2. 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.

4
4 Graphing y = nx 2 The Graph of y = x 2 is a Parabola Lesson You can find out what the graph of y = x 2 looks like by plotting points. Examples 1 and 2 show you how to plot a graph of y = x 2 for positive and negative values of x.

5
5 Graphing y = nx 2 Lesson Example 1 Solution follows… Plot the graph of y = x 2 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 x 456 y (= x 2 ) 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. y x

6
6 Graphing y = nx 2 Example 2 Solution follows… Lesson Plot the graph of y = x 2 for values of x between –6 and 6. Solution The table of values looks like this: –3–2–10 x –4–5–6 y (= x 2 ) This kind of curve is called a parabola –2–4–6 And the curve looks like this. y x

7
7 Graphing y = nx 2 Guided Practice Solution follows… Lesson Which of the following points are on the graph of y = x 2 ? (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 = x 2 whose x -coordinate is shown –10 4. – (1, 1), (–1, 1), (2, 4), (3, 9), and (5, 25)

8
8 Graphing y = nx 2 Guided Practice Solution follows… Lesson In Exercises 6–9, calculate the two possible x -coordinates of the points on the graph of y = x 2 whose y -coordinate is shown and –45 and –5 7 and –7 and –

9
9 Graphing y = nx 2 You Can Use a Graph to Solve an Equation Lesson 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.

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

11
11 Graphing y = nx 2 Example 3 Lesson Using the graph of y = x 2 in Example 2, solve x 2 = 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).

12
12 Graphing y = nx 2 Guided Practice Solution follows… Lesson Use the graph of y = x 2 shown below to solve the equations in Exercises 10– x 2 = x 2 = x 2 = x 2 = –2–4–6 x = 4 or x = –4 x = 5 or x = –5 x = 3.2 or x = –3.2 (approximately) x = 5.5 or x = –5.5 (approximately) x = –5.5 x = 5.5 x = –4 x = 4 x = –5 x = 5 x = –3.2 x = 3.2 y x

13
13 Graphing y = nx 2 The Graph of y = nx 2 is Also a Parabola Lesson The graph of y = x 2 is y = nx 2 where n = 1. It has the U shape of a parabola. Other values of n give graphs that look very similar.

14
14 Graphing y = nx 2 Example 4 Solution follows… Lesson Plot the graphs of the following equations for values of x between –5 and 5. a) y = 2 x 2 b) y = 3 x 2 c) y = 4 x 2 d) y = x Solution The best place to start is with a table of values, just like before. All these equations are of the form y = nx 2, for different values of n (2 then 3 then 4 then ). 1 2 Solution continues…

15
15 You then need to plot the y -values in each colored column against the x ‑ values in the first column. Graphing y = nx 2 Example 4 Lesson Plot the graphs of the following equations for values of x between –5 and 5. a) y = 2 x 2 b) y = 3 x 2 c) y = 4 x 2 d) y = x Solution (continued) The table on the right shows values for parts a)–d). x 2x22x2 0 1 and –1 2 and –2 3 and –3 4 and –4 5 and – x23x x24x ½ x2½ x Solution continues…

16
16 Graphing y = nx 2 Example 4 Lesson –2–4231–1– –5 Solution (continued) x 2x22x2 0 1 and –1 2 and –2 3 and –3 4 and –4 5 and – x 0 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 3x23x x 0 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 4x24x x 0 1 and –1 2 and –2 3 and –3 4 and –4 5 and –5 ½ x2½ x y = 4 x 2 y = 3 x 2 y = 2 x 2 y = x Increasing values of n Decreasing values of n ( n = 4) ( n = 3) ( n = 2) ( n = ½) y x

17
17 Graphing y = nx 2 Lesson 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. In fact, this is a general rule — if n is positive, the graph of y = nx 2 will always be a “u-shaped” parabola with its vertex at the origin –2–4231–1– –5 y = 4 x 2 y = 3 x 2 y = 2 x 2 y = x y x

18
18 Graphing y = nx 2 Lesson In Example 4, the graph of y = 4 x 2 had the steepest parabola, while the graph of y = ½ x 2 was the least steep. Also, the greater the value of n, the steeper the parabola will be –2–4231–1– –5 y = 4 x 2 y = 3 x 2 y = 2 x 2 y = x y x

19
19 Graphing y = nx 2 Guided Practice Solution follows… Lesson For Exercises 14–17, draw on the same axes the graph of each of the given equations. 14. y = 5 x y = x y = 10 x y = x –2– y = 5 x 2 y = 10 x 2 y = x y x

20
20 Graphing y = nx 2 Guided Practice Solution follows… Lesson In Exercises 18–23, use the graphs from Example 4 to solve the given equations x 2 = x 2 = x 2 = x 2 = x 2 = x 2 = x 3.2 or x –3.2 x 1.9 or x –1.9 x 4.8 or x –4.8 x 2.9 or x –2.9 x 4.5 or x –4.5 x 4.6 or x – –2– y = 4 x 2 y = 3 x 2 y = 2 x 2 y = x y x

21
21 Graphing y = nx 2 Independent Practice Solution follows… Lesson Using a table of values, plot the graphs of the equations in Exercises 1–3 for values of x between –4 and y = 1.5 x 2 2. y = 5 x 2 3. y = x –2–4231–1– y = 1.5 x 2 y = 5 x 2 y = x y x

22
22 Graphing y = nx 2 Independent Practice Solution follows… Lesson 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 = 2.5 x 2 5. y = 6 x 2 6. y = x –2–4231–1– y = 1.5 x 2 y = 5 x 2 y = x y = 6 x 2 y = 2.5 x 2 y = x y x

23
23 Graphing y = nx 2 Independent Practice Solution follows… Lesson If s is the length of a square’s sides, then a formula for its area, A, is A = s 2. Plot a graph of A against s, for values of s up to 10.

24
24 Graphing y = nx 2 Independent Practice Solution follows… Lesson On a graph of y = x 2, what is the y -coordinate when x = 10 3 ? For Exercises 9–12, find the y -coordinate of the point on the graph of y = x 2 for each given value of x. 9. x = 10 –1 10. x = 10 –4 11. x = 12. x = 10 –2 10 –

25
25 Graphing y = nx 2 Independent Practice Solution follows… Lesson For Exercises 13–15, find the x -coordinates of the point on the y = x 2 graph for each given value of y. 13. y = y = 10 –6 15. y = and –10 10 –3 and –10 –3 2 4 and –2 4

26
26 Graphing y = nx 2 Round Up Lesson In this Lesson you’ve looked at graphs of the form y = nx 2, where n is positive. Remember that, because in the next Lesson you’re going to look at graphs of the same form where n is negative. 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.

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google