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Example 1: Real-World Example: Estimate Function Values Five-Minute Check Then/Now New Vocabulary Example 1: Real-World Example: Estimate Function Values Example 2: Find Domain and Range Example 3: Find y-Intercepts Example 4: Find Zeros Key Concept: Tests for Symmetry Example 5: Test for Symmetry Key Concept: Even and Odd Functions Example 6: Identify Even and Odd Functions Lesson Menu
Determine whether 2y + 5x = 7 represents y as a function of x. A. y is a function of x. B. y is not a function of x. 5–Minute Check 1
Determine whether the graph represents y as a function of x. A. y is a function of x. B. y is not a function of x. 5–Minute Check 2
Find f (–2) if f (x) = 6 – x 2. A. 2 B. 4 C. 8 D. 10 5–Minute Check 3
Find A. 3a – 2 B. C. D. 5–Minute Check 4
State the domain of . A. B. (–3, 3) C. D. 5–Minute Check 5
Determine whether 2y + 5x = 7 represents y as a function of x. A. y is a function of x. B. y is not a function of x. 5–Minute Check 1
Determine whether the graph represents y as a function of x. A. y is a function of x. B. y is not a function of x. 5–Minute Check 2
Find f (–2) if f (x) = 6 – x 2. A. 2 B. 4 C. 8 D. 10 5–Minute Check 3
Find A. 3a – 2 B. C. D. 5–Minute Check 4
State the domain of . A. B. (–3, 3) C. D. 5–Minute Check 5
You identified functions. (Lesson 1-1) Use graphs of functions to estimate function values and find domains, ranges, y-intercepts, and zeros of functions. Explore symmetries of graphs, and identify even and odd functions. Then/Now
zeros roots line symmetry point symmetry even function odd function Vocabulary
Estimate Function Values A. ADVERTISING The function f (x) = –5x 2 + 50x approximates the profit at a toy company, where x represents marketing costs and f (x) represents profit. Both costs and profits are measured in tens of thousands of dollars. Use the graph to estimate the profit when marketing costs are $30,000. Confirm your estimate algebraically. Example 1
The graphical estimate of about $1,000,000 is reasonable. Estimate Function Values $30,000 is three ten thousands. The function value at x = 3 appears to be about 100 ten thousands, so the total profit was about $1,000,000. To confirm this estimate algebraically, find f(3). f(3) = -5(3)2 + 50(3) = 105, or about $1,050,000. The graphical estimate of about $1,000,000 is reasonable. Answer: $1,050,000 Example 1
Estimate Function Values B. ADVERTISING The function f (x) = –5x 2 + 50x approximates the profit at a toy company, where x represents marketing costs and f (x) represents profit. Both costs and profits are measured in tens of thousands of dollars. Use the graph to estimate marketing costs when the profit is $1,250,000. Confirm your estimate algebraically. Example 1
Estimate Function Values $1,250,000 is 125 ten thousands. The value of the function appears to reach 125 ten thousands for an x-value of about 5. This represents an estimate of 5 ● $10,000 or $50,000. To confirm algebraically, find f(5). f(5) = -5(5)2 + 50(5) = 125, or about $1,250,000. The graphical estimate that marketing costs are $50,000 when the profit is $1,250,000 is reasonable. Answer: $50,000 Example 1
PROFIT A-Z Toy Boat Company found the average price of its boats over a six month period. The average price for each boat can be represented by the polynomial function p (x) = –0.325x3 + 1.5x2 + 22, where x is the month, and 0 < x ≤ 6. Use the graph to estimate the average price of a boat in the fourth month. Confirm you estimate algebraically. A. $25 B. $23 C. $22 D. $20 Example 1
PROFIT A-Z Toy Boat Company found the average price of its boats over a six month period. The average price for each boat can be represented by the polynomial function p (x) = –0.325x3 + 1.5x2 + 22, where x is the month, and 0 < x ≤ 6. Use the graph to estimate the average price of a boat in the fourth month. Confirm you estimate algebraically. A. $25 B. $23 C. $22 D. $20 Example 1
Use the graph of f to find the domain and range of the function. Find Domain and Range Use the graph of f to find the domain and range of the function. Example 2
The domain of f is . In set-builder notation, the domain is . Find Domain and Range Domain The dot at (3, -3) indicates that the domain of f ends at 3 and includes 3. The arrow on the left side indicates that the graph will continue without bound. The domain of f is . In set-builder notation, the domain is . Range The graph does not extend above y = 2, but f (x) decreases without bound for smaller and smaller values of x. So the range of f is . Example 2
Find Domain and Range Answer: D: R: Example 2
Use the graph of f to find the domain and range of the function. A. Domain: Range: B. Domain: Range: C. Domain: Range: D. Domain: Range: Example 2
Use the graph of f to find the domain and range of the function. A. Domain: Range: B. Domain: Range: C. Domain: Range: D. Domain: Range: Example 2
Find y-Intercepts A. Use the graph of the function f (x) = x 2 – 4x + 4 to approximate its y-intercept. Then find the y-intercept algebraically. Example 3
Find y-Intercepts Estimate Graphically It appears that f (x) intersects the y-axis at approximately (0, 4), so the y-intercept is about 4. Solve Algebraically Find f (0). f (0) = (0)2 – 4(0) + 4 = 4. The y-intercept is 4. Answer: 4 Example 3
Find y-Intercepts B. Use the graph of the function g (x) =│x + 2│– 3 to approximate its y-intercept. Then find the y-intercept algebraically. Example 3
Find y-Intercepts Estimate Graphically g (x) intersects the y-axis at approximately (0, -1), so the y-intercept is about -1. Solve Algebraically Find g (0). g (0) = |0 + 2| – 3 or –1 The y-intercept is –1. Answer: -1 Example 3
Use the graph of the function to approximate its y-intercept Use the graph of the function to approximate its y-intercept. Then find the y-intercept algebraically. A. –1; f (0) = –1 B. 0; f (0) = 0 C. 1; f (0) = 1 D. 2; f (0) = 2 Example 3
Use the graph of the function to approximate its y-intercept Use the graph of the function to approximate its y-intercept. Then find the y-intercept algebraically. A. –1; f (0) = –1 B. 0; f (0) = 0 C. 1; f (0) = 1 D. 2; f (0) = 2 Example 3
Find Zeros Use the graph of f (x) = x 3 – x to approximate its zero(s). Then find its zero(s) algebraically. Example 4
The x-intercepts appear to be at about -1, 0, and 1. Find Zeros Estimate Graphically The x-intercepts appear to be at about -1, 0, and 1. Solve Algebraically x 3 – x = 0 Let f (x) = 0. x(x 2 – 1) = 0 Factor. x(x – 1)(x + 1) = 0 Factor. x = 0 or x – 1 = 0 or x + 1 = 0 Zero Product Property x = 0 x = 1 x = -1 Solve for x. The zeros of f are 0, 1, and -1. Answer: -1, 0, 1 Example 4
Use the graph of to approximate its zero(s) Use the graph of to approximate its zero(s). Then find its zero(s) algebraically. A. –2.5 B. –1 C. 5 D. 9 Example 4
Use the graph of to approximate its zero(s) Use the graph of to approximate its zero(s). Then find its zero(s) algebraically. A. –2.5 B. –1 C. 5 D. 9 Example 4
Key Concept 1
Test for Symmetry A. Use the graph of the equation y = x 2 + 2 to test for symmetry with respect to the x-axis, the y-axis, and the origin. Support the answer numerically. Then confirm algebraically. Example 5
A table of values supports this conjecture. Test for Symmetry Analyze Graphically The graph appears to be symmetric with respect to the y-axis because for every point (x, y) on the graph, there is a point (-x, y). Support Numerically A table of values supports this conjecture. Example 5
Confirm Algebraically Test for Symmetry Confirm Algebraically Because x2 + 2 is equivalent to (-x)2 + 2, the graph is symmetric with respect to the y-axis. Answer: symmetric with respect to the y-axis Example 5
Test for Symmetry B. Use the graph of the equation xy = –6 to test for symmetry with respect to the x-axis, the y-axis, and the origin. Support the answer numerically. Then confirm algebraically. Example 5
A table of values supports this conjecture. Test for Symmetry Analyze Graphically The graph appears to be symmetric with respect to the origin because for every point (x, y) on the graph, there is a point (-x, -y). Support Numerically A table of values supports this conjecture. Example 5
Confirm Algebraically Test for Symmetry Confirm Algebraically Because (-x)( -y) = -6 is equivalent to (x)(y) = -6, the graph is symmetric with respect to the origin. Answer: symmetric with respect to the origin Example 5
A. symmetric with respect to the x-axis Use the graph of the equation y = –x 3 to test for symmetry with respect to the x-axis, the y-axis, and the origin. Support the answer numerically. Then confirm algebraically. A. symmetric with respect to the x-axis B. symmetric with respect to the y-axis symmetric with respect to the origin D. not symmetric with respect to the x-axis, y-axis, or the origin Example 5
A. symmetric with respect to the x-axis Use the graph of the equation y = –x 3 to test for symmetry with respect to the x-axis, the y-axis, and the origin. Support the answer numerically. Then confirm algebraically. A. symmetric with respect to the x-axis B. symmetric with respect to the y-axis symmetric with respect to the origin D. not symmetric with respect to the x-axis, y-axis, or the origin Example 5
Key Concept 2
Identify Even and Odd Functions A. Graph the function f (x) = x 2 – 4x + 4 using a graphing calculator. Analyze the graph to determine whether the function is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function. Example 6
f (-x) = (-x) 2 – 4(-x) + 4 Substitute -x for x. Identify Even and Odd Functions It appears that the graph of the function is neither symmetric with respect to the y-axis nor to the origin. Test this conjecture algebraically. f (-x) = (-x) 2 – 4(-x) + 4 Substitute -x for x. = x 2 + 4x + 4 Simplify. Since –f (x) = -x 2 + 4x - 4, the function is neither even nor odd because f (-x) ≠ f (x) or –f (x). Answer: neither Example 6
Identify Even and Odd Functions B. Graph the function f (x) = x 2 – 4 using a graphing calculator. Analyze the graph to determine whether the function is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function. Example 6
f (-x) = (-x)2 – 4 Substitute -x for x. = x 2 - 4 Simplify. Identify Even and Odd Functions From the graph, it appears that the function is symmetric with respect to the y-axis. Test this conjecture algebraically. f (-x) = (-x)2 – 4 Substitute -x for x. = x 2 - 4 Simplify. = f (x) Original function f (x) = x 2 – 4 The function is even because f (-x) = f (x). Answer: even; symmetric with respect to the y-axis Example 6
Identify Even and Odd Functions C. Graph the function f (x) = x 3 – 3x 2 – x + 3 using a graphing calculator. Analyze the graph to determine whether the function is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function. Example 6
f (–x) = (–x) 3 – 3(–x)2 – (–x) + 3 Substitute –x for x. Identify Even and Odd Functions From the graph, it appears that the function is neither symmetric with respect to the y-axis nor to the origin. Test this conjecture algebraically. f (–x) = (–x) 3 – 3(–x)2 – (–x) + 3 Substitute –x for x. = –x 3 – 3x 2 + x + 3 Simplify. Because –f (x) = –x 3 + 3x 2 + x – 3, the function is neither even nor odd because f (–x) ≠ f (x) or –f (x). Answer: neither Example 6
A. odd; symmetric with respect to the origin Graph the function f (x) = x 4 – 8 using a graphing calculator. Analyze the graph to determine whether the graph is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function. A. odd; symmetric with respect to the origin B. even; symmetric with respect to the y-axis C. neither even nor odd Example 6
A. odd; symmetric with respect to the origin Graph the function f (x) = x 4 – 8 using a graphing calculator. Analyze the graph to determine whether the graph is even, odd, or neither. Confirm algebraically. If even or odd, describe the symmetry of the graph of the function. A. odd; symmetric with respect to the origin B. even; symmetric with respect to the y-axis C. neither even nor odd Example 6
That’s a wrap Practice 1-2 HW # 1- 41 (odd), 107-110 Get ready for the 1-1, 1-2 Quiz!