Toolkit of Basic Function Families Solutions

Transcription

1 I. Linear Functions a. Domain and range: All real numbers unless a = 0, then the range is onl b. b. Graph patterns: Straight lines with slope a and -intercept (0, b) c. Table patterns: Constant additive rate of change so that _ Δ Δx = a or f (x + 1) - f (x) = a and the pair (0, b) appears in the table f(x) = 0.5x e. Smmetries: A line has 180 rotational smmetr about an point on the line. Lines are also reflection smmetric across an reflection line that is perpendicular to the original line. DIFFERENTIATION You ma wish to push some students to think about the following. Technicall, a line has translation smmetr when the translation is parallel to the line and glide reflection smmetr for a parallel translation and line reflection across itself. f. Asmptotes: None - x - g(x) = -x + 1 g. Increasing/decreasing patterns: If the slope is positive, the linear function is increasing. If the slope is negative, the function is decreasing. II. Exponential Functions Range: All positive real numbers when a > 0; All negative real numbers when a < 0 Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. b. Graph patterns: Curves that are increasing at an increasing rate for b > 1 and decreasing at a decreasing rate for b < 1, with -intercept (0, a) in both cases c. Table patterns: Constant multiplicative rate of change so f (x + 1) that _ = b and pair (0, a) is in the table f (x) e. Smmetries: None f. Asmptotes: The line = 0 g. Increasing/decreasing patterns: If the base b of the exponent is greater than 1, the function is increasing. If 0 < b < 1, the function is decreasing. f(x) = (1.5 x ) 6 g(x) = (0.8 x ) - x Student Master use with page 1 Unit 1 11

2 III. Power Functions Range: Positive numbers when a > 0 and n is even; negative numbers when a < 0 and n is even; all real numbers when n is odd b. Graph patterns: When n is even, power functions are U-shaped and curved up when a > 0 and U-shaped and curved down when a < 0. When n is odd, the shape is increasing with slow changes near the origin or decreasing with slow changes near the origin. Sample graphs: n is even n is odd f(x) = x 3 1 f(x) = x x x -1 g(x) = x -1 g(x) = -x c. Table patterns: When n is even, table values decrease to zero and then increase, or increase to zero and then decrease. When n is odd, table values either continuousl increase or continuousl decrease for all power functions. d. Maximum/minimum points: When n is odd, there are no maximum or minimum points. When n is even and a > 0, the minimum point is at (0, 0), while there is no maximum point. When n is even and a < 0, the maximum point is at (0, 0), while there is no minimum point. e. Smmetries: Reflection smmetr across the -axis when n is even; 180 rotational smmetr about the origin when n is odd f. Asmptotes: None g. Increasing/decreasing patterns: For odd power functions, the function is increasing if a > 0 and decreasing if a < 0. For even power functions, when a > 0, the function decreases on the interval (-, 0] and increases on [0, ). When a < 0, the function increases on (-, 0] and decreases on [0, ). Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. 1 Unit 1 Student Master use with page 1

3 IV. Inverse Variation Functions except 0 Range: All real numbers except 0, when n is odd; all positive real numbers when n is even and a > 0; all negative real numbers when n is even and a < 0 b. Graph patterns: For n odd and a > 0, the values are in Quadrants I and III. In Quadrant I, the graph is a curve that approaches the -axis from above x = 0 and approaches the x-axis when x is large. A similar pattern exists for the section of the graph in Quadrant III. In fact, this part of the curve is a reflection of the curve in Quadrant I across = -x. For n odd and a < 0, the two parts of the graph are in Quadrants II and IV. For n even and a > 0, the graph is in Quadrants I and II and both branches open up. For n even and a < 0, the graph is in Quadrants III and IV and both branches open down. Sample graphs: n is odd n is even f(x) = 5 x f(x) = 5 x - x - x - - Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. c. Table patterns: For n odd and a > 0, values of are negative and decreasing at an increasing rate as x approaches 0 from below; values of are positive and decreasing at a decreasing rate as x increases above 0. For n odd and a < 0, the values of are positive and increase at an increasing rate as x approaches 0 from below; values of are negative and increase at a decreasing rate as x increases above 0. For n even and a > 0, values of are positive and increasing at an increasing rate as x approaches 0 from below; values of are positive and decreasing at a decreasing rate as x increases above 0. For n even and a < 0, the values of are negative and decreasing at an increasing rate as x approaches 0 from below; values of are negative and increasing at a decreasing rate as x increases above 0. Student Master use with page 1 Unit 1 13

4 e. Smmetries: 180 rotational smmetr about the origin when n is odd; reflection smmetr across the -axis when n is even f. Asmptotes: Lines = 0 and x = 0 g. Increasing/decreasing patterns: When n is odd and a > 0, both branches will be decreasing; when a < 0, both branches will be increasing. When n is even and a > 0, the branch in Quadrant II will be increasing and the branch in Quadrant I will be decreasing; when a < 0, the branch in Quadrant III will be decreasing and the branch in Quadrant IV will be increasing. V. Quadratic Functions Range: All real numbers greater than or equal to the minimum value (if a > 0) or less than or equal to the maximum value (if a < 0). b. Graph patterns: Parabolas that are curved up when a > 0 and curved down when a < 0; -intercept at (0, c); vertex at ( _-b a, c - _ a) b c. Table patterns: For a > 0, values of decrease at a decreasing rate to the minimum when x = _-b and then increase at an a increasing rate. For a < 0, values of increase at a decreasing rate to the maximum when x = _-b and then decrease at an a increasing rate. The pair (0, c) is in the table. g(x) = 0.5x -x - 1 d. Maximum/minimum points: If a > 0, the graph will have a minimum point and no maximum point. If a < 0, the graph will have a maximum point and no minimum point. Minimum or maximum values occur when x = _-b a. e. Smmetries: Reflection smmetr across the vertical line containing the vertex, x = _-b a f. Asmptotes: None g. Increasing/decreasing patterns: When a > 0, the function will be decreasing from left to right until the minimum point is reached and then increasing thereafter. When a < 0, the function will be increasing from left to right until the maximum point is reached and then decreasing thereafter. - x - f(x) = 0.5x x + 1 Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. 1 Unit 1 Student Master use with page 1

5 VI. Circular Functions Range: - a a b. Graph patterns: Shown at the right c. Table patterns: Values of the dependent variable oscillate between maximum and minimum values (see range) with period π. c(x) = cos x s(x) = sin x d. Maximum/minimum points: Both sine and cosine functions have maximum of a and minimum of - a. - e. Smmetries: The sine and cosine functions have horizontal translation smmetr (±πn where n is a positive integer). The also have reflection smmetr across vertical lines that contain maximum or minimum points of the functions. In addition, the sine and cosine functions have 180 rotational smmetr about an x-intercept. f. Asmptotes: None - 3π -π - π π π 3π x g. Increasing/decreasing patterns: The sine function will be decreasing on the intervals [ π_ + πn, _ 3π + πn ] for integers n. It will be increasing on the intervals [ - π_ + πn, π_ + πn ]. The cosine function will be decreasing on the intervals [0 + πn, π + πn]. It will be increasing on the intervals [π + πn, π + πn]. Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. Student Master use with page 1 Unit 1 15

6 VII. Common Logarithmic Functions greater than 0 Range: All real numbers b. Graph patterns: When a > 0, as x gets larger, the graph rises quickl between x = 0 and x = 1. The x-intercept is at x = 1. As x gets larger than 1, the graph rises more slowl. When a < 0, the graph between x = 0 and x = 1 decreases rapidl and then continues to decrease slowl as x gets larger. c. Table patterns: When a > 0, as x gets larger, f (x) quickl increases from negative values to zero. Then the values increase ver slowl. When a < 0, f (x) quickl decreases from positive values to 0. Then the values decrease ver slowl f(x) = log x 6 8 g(x) = -log x x - e. Smmetries: None f. Asmptotes: Line x = 0 g. Increasing/decreasing patterns: For a > 0, the logarithmic function increases on its domain (0, ). For a < 0, the logarithmic function decreases on its domain (0, ). Copright McGraw-Hill Education. Permission is granted to reproduce for classroom use. 16 Unit 1 Student Master use with page 1