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Lesson Menu Five-Minute Check Then/Now New Vocabulary Key Concept:Limits Key Concept:Types of Discontinuity Concept Summary: Continuity Test Example 1:Identify a Point of Continuity Example 2:Identify a Point of Discontinuity Key Concept:Intermediate Value Theorem Example 3:Approximate Zeros Example 4:Graphs that Approach Infinity Example 5:Graphs that Approach a Specific Value Example 6:Real-World Example: Apply End Behavior

5–Minute Check 1 Use the graph of f (x) to find the domain and range of the function. A.D =, R = B.D =, R = [–5, 5] C.D = (–3, 4), R = (–5, 5) D. D = [–3, 4], R = [–5, 5]

5–Minute Check 2 Use the graph of f (x) to find the y-intercept and zeros. Then find these values algebraically. A.y-intercept = 9, zeros: 2 and 3 B.y-intercept = 8, zeros: 1.5 and 3 C.y-intercept = 9, zeros: 1.5 and 3 D.y-intercept = 8, zero: –1

5–Minute Check 3 Use the graph of y = –x 2 to test for symmetry with respect to the x-axis, y-axis, and the origin. A.y-axis B.x-axis C.origin D.x- and y-axis

Then/Now You found domain and range using the graph of a function. (Lesson 1-2) Use limits to determine the continuity of a function, and apply the Intermediate Value Theorem to continuous functions. Use limits to describe end behavior of functions.

Vocabulary continuous function limit discontinuous function infinite discontinuity jump discontinuity removable discontinuity nonremovable discontinuity end behavior

Key Concept 1

Key Concept 2

Concept Summary 1

Example 1 Identify a Point of Continuity Check the three conditions in the continuity test. Determine whether is continuous at. Justify using the continuity test. Because, the function is defined at 1.Does exist?

Example 1 Identify a Point of Continuity 2.Does exist? Construct a table that shows values of f(x) approaching from the left and from the right. The pattern of outputs suggests that as the value of x gets close to from the left and from the right, f(x) gets closer to. So we estimate that.

Example 1 Identify a Point of Continuity 3.Does ? Because is estimated to be and we conclude that f (x) is continuous at. The graph of f (x) below supports this conclusion.

Example 1 Identify a Point of Continuity Answer: exists. 3.. f (x) is continuous at.

Example 1 Determine whether the function f (x) = x 2 + 2x – 3 is continuous at x = 1. Justify using the continuity test. A.continuous; f (1) B.Discontinuous; the function is undefined at x = 1 because does not exist.

Example 2 Identify a Point of Discontinuity A. Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1.Because, is undefined, f (1) does not exist.

Example 2 Identify a Point of Discontinuity 2.Investigate function values close to f(1). The pattern of outputs suggests that for values of x approaching 1 from the left, f (x) becomes increasingly more negative. For values of x approaching 1 from the right, f (x) becomes increasing more positive. Therefore, does not exist.

Example 2 Identify a Point of Discontinuity Answer: f (x) has an infinite discontinuity at x = 1. 3.Because f (x) decreases without bound as x approaches 1 from the left and f (x) increases without bound as x approaches 1 from the right, f (x) has infinite discontinuity at x = 1. The graph of f (x) supports this conclusion.

Example 2 Identify a Point of Discontinuity B. Determine whether the function is continuous at x = 2. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1.Because, is undefined, f (2) does not exist. Therefore f (x) is discontinuous at x = 2.

Example 2 Identify a Point of Discontinuity 2.Investigate function values close to f (2). The pattern of outputs suggests that f (x) approaches 0.25 as x approaches 2 from each side, so.

Example 2 Identify a Point of Discontinuity 3.Because exists, but f (2) is undefined, f (x) has a removable discontinuity at x = 2. The graph of f (x) supports this conclusion. Answer: f (x) is not continuous at x = 2, with a removable discontinuity.

Example 2 A.f (x) is continuous at x = 1. B.infinite discontinuity C.jump discontinuity D.removable discontinuity Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

Key Concept 3

Example 3 Approximate Zeros Investigate function values on the interval [  2, 2]. A. Determine between which consecutive integers the real zeros of are located on the interval [–2, 2].

Example 3 Approximate Zeros Answer: There are two zeros on the interval, –1 < x < 0 and 1 < x < 2. Because f (  1) is positive and f (0) is negative, by the Location Principle, f (x) has a zero between  1 and 0. The value of f (x) also changes sign for [1,2]. This indicates the existence of real zeros in each of these intervals. The graph of f (x) supports this conclusion.

Example 3 Approximate Zeros B. Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x + 5 are located on the interval [–2, 2]. Investigate function values on the interval [–2, 2].

Example 3 Answer:–2 < x < –1. Approximate Zeros Because f (  2) is negative and f (–1) is positive, by the Location Principle, f (x) has a zero between –2 and –1. This indicates the existence of real zeros on this interval. The graph of f (x) supports this conclusion.

Example 3 A. Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x 2 – x – 1 are located on the interval [–4, 4]. A.–1 < x < 0 B.–3 < x < –2 and –1 < x < 0 C.–3 < x < –2 and 0 < x < 1 D.–3 < x < –2, –1 < x < 0, and 0 < x < 1

Example 3 B. Determine between which consecutive integers the real zeros of f (x) = 3x 3 – 2x are located on the interval [–2, 2]. A.–2 < x < –1 B.–1 < x < 0 C.0 < x < 1 D.1 < x < 2

Example 4 Graphs that Approach Infinity Use the graph of f(x) = x 3 – x 2 – 4x + 4 to describe its end behavior. Support the conjecture numerically.

Example 4 Graphs that Approach Infinity Analyze Graphically Support Numerically Construct a table of values to investigate function values as |x| increases. That is, investigate the value of f (x) as the value of x becomes greater and greater or more and more negative. In the graph of f (x), it appears that and

Example 4 Graphs that Approach Infinity The pattern of output suggests that as x approaches –∞, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞. Answer:

Example 4 Use the graph of f (x) = x 3 + x 2 – 2x + 1 to describe its end behavior. Support the conjecture numerically. A. B. C. D.

Example 5 Graphs that Approach a Specific Value Use the graph of to describe its end behavior. Support the conjecture numerically.

Example 5 Graphs that Approach a Specific Value Analyze Graphically Support Numerically In the graph of f (x), it appears that. As. As. This supports our conjecture.

Example 5 Graphs that Approach a Specific Value Answer:

Example 5 Use the graph of to describe its end behavior. Support the conjecture numerically. A. B. C. D.

Example 6 Apply End Behavior PHYSICS The symmetric energy function is. If the y-value is held constant, what happens to the value of symmetric energy when the x-value approaches negative infinity? We are asked to describe the end behavior of E (x) for small values of x when y is held constant. That is, we are asked to find.

Example 6 Apply End Behavior Because y is a constant value, for decreasing values of x, the fraction will become larger and larger, so. Therefore, as the x-value gets smaller and smaller, the symmetric energy approaches the value Answer:

Example 6 PHYSICS The illumination E of a light bulb is given by, where I is the intensity and d is the distance in meters to the light bulb. If the intensity of a 100-watt bulb, measured in candelas (cd), is 130 cd, what happens to the value of E when the d-value approaches infinity? A. B. C. D.

End of the Lesson