1. WARM UP If f(x) = x, find f(3) If f(x) = x, find f(0) If f(x) = x, find f(-3) If g(x) = 2, find g(3) If h(x) = x, find h(25) What property of real numbers is illustrated by 3(x + 5) = 3(5 + x)? 9 0 9 8 5 Commutative property of addition

2. IDENTIFYING FUNCTIONS FROM GRAPHICAL PATTERNS

3. OBJECTIVES Given the graph of a function, know whether the function is exponential, power, quadratic, or linear and find the particular equation algebraically.

4. KEY TERMS & CONCEPTS Slope-intercept form Point-slope form Slope Vertex Vertex form Parabola Proportionality constant Directly proportional Inversely proportional Base-10 exponential function Natural (base-e exponential function

5. LINEAR & CONSTANT FUNCTIONS General equation: y = ax + b (often written y = mx + b), where a (or m) and b stand for constants and the domains is all real numbers. This equation is in the slope-intercept form because a (or m) gives the slope and b gives the y-intercept. If a = 0, then y = b; this is a constant function. Parent function: y = x

6. LINEAR & CONSTANT FUNCTIONS Transformed funtion: called the point-slope form because the graph contains and has a slope a. The slope, a is the vertical dilation; is the vertical translation; and is the horizontal translation. Note that the point-slope form can also be written, where the coordinates of the fixed point both appear with a – sign. The form expresses y explicitly in terms of x and thus is easier to enter into your grapher.

7. LINEAR & CONSTANT FUNCTIONS Graphical properties: The graph is a straight line. The parent function is shown in the left graph, the slope- intercept form is shown in the middle graph, and the point slope form is shown in the right graph. Verbally: For the slope-intercept form: “Start at b on the y-axis, run x, and rise ax.” For point-slope form: “Start at, run, and rise.”

8. QUADRATIC FUNCTIONS General equation: y = ax + bx + c, where a ≠ 0, a, b, and c stand for constants, and the domain is all real numbers. Parent function: y = x where the vertex is at the origin. Transformed function: called the vertex form, with vertex at (h, k). The value k is the vertical translation, h is the horizontal translation, and a is the vertical dilation. Vertex form can also be written,, expressing y explicitly in terms of x is easier to enter into your grapher.

9. QUADRATIC FUNCTIONS Graphical properties: The graph is a parabola. Greek for “along the path of a ball”) as shown in the graphs below. The graph is concave up if a > 0 and concave down if a < 0.

10. POWER FUNCTIONS General equation: y = ax where a and b stand for nonzero constants. If b > 0, then the domain can be all real numbers. If b < 0, then the domain excludes x = 0 to avoid division by zero. If b is not an integer, then the domain excludes negative numbers to avoid roots of negative numbers. The domain is also restricted to nonnegative numbers in most applications. Parent function: y = x

11. POWER FUNCTIONS Graphical properties: The graphs are for different values of b. In all three cases a > 0. The shape and concavity of the graph depend on the value of b. The graph contains the origin if b > 0; it has the axes as asymptotes if b 0; it is decreasing if b 0, concave up means y is increasing at an increasing rate, and concave down means it is increasing at a decreasing rate.

12. POWER FUNCTIONS Verbally: y = ax,” if b > 0, y varies with the b power of x, or y is directly proportional to the b power of x; if b < 0, then y varies inversely with the b power of x, or y is inversely proportional to the b power ox x.” The constant a is the proportionality constant. Translated function: y =d + a(x – c), where c and d are the horizontal and vertical translation, respectively. Compare the translated form with for linear functions for quadratic functions For sinusoidal functions Unless otherwise stated, “power function” will imply untranslated form, y = ax

13. EXPONENTIAL FUNCTIONS General equation: y = ab, where a and be are constants and a ≠ 0, b > 0, b ≠ 1, and the domain is all real numbers. Parent function: y = b Verbally : In the equation y = ab, “y varies exponential with x.” Translated function: y = ab + c, where the asymptote is the line y = c. Unless otherwise stated, “exponential function” will imply the untranslated form y = ab

14. EXPONENTIAL FUNCTIONS Graphical properties: The graph shows exponential for different values of a and b. The constant a is the y-intercept. The function is increasing if b > 1 and decreasing if 0 0). If a 0 and concave down if a < 0.

15. SPECIAL EXPONENTIAL FUNCTIONS Mathematicians usually use one of two particular constants for the base of an exponential function: either 10, which is the base of the decimal system, or the naturally occurring number e, which equals 2.78128… To make the equation more general, multiply the variable in the exponent by a constant. The (untranslated) general equations are given below. DEFINITION: base-10 exponential function natural (base-e) exponential function where a and b are constants and the domain is all real numbers.

16. SPECIAL EXPONENTIAL FUNCTIONS Note: The equations of these two functions can be generalized by incorporating translations in the x- and y- directions. You’ll get and Base-e exponential functions have an advantage when you study calculus because the rate of change of is equal to.

17. EXAMPLE 1 For the function graphed: a.Identify the kind of function it is b.On what interval or intervals is the function increasing or decreasing? Which way is the graph concave up or down? c. From your experience, describe something in the real world that a function with this shape graph could model. d.Find the particular equation of the function, given points (5, 19) and (10, 6) are on the graph. e.Confirm by plotting that your equation give the same graph.

18. WARM UP Power functions and exponential functions both have exponents. What major algebraic difference distinguishes these two types of functions? Write a sentence giving the origin of the word concave and explaining how the word applies to the graph of a function Explain why the reciprocal function f(x) = 1/x is also a power function. What are the parent functions of the following graphs: Linear quadratic power exponential Power function: exponent is constant, independent variable is the base, Exponential is the opposite. “ The concave side of a curved portion of a graph is the inside of that curve.” It is equal to x

19. SOLUTION a.Because the graph is a straight line, the function is linear. b. The function is decreasing over its entire domain, and the graph is not concave in either direction. c. The function could model anything that decreases at a constant rate. The Quadrant 1 part of the function could model the number of pages of history text you have left to read as a function of the number of minutes you have been reading.

20. SOLUTION CONTINUED d. f(x) = ax + b Write the general equation. Use f(x) as shown on the graph, and use a for the slope. 19 = 5a + b 6 = 10a + b -13 = 5a + b Substitute the given values of x and y into the equation of f. Substitute the first equation from the second to eliminate b. 6 = 10(-2.6) + 32 Substitute -2.6 for a in one of the equations. f(x) = 2.6x + 32 Write the particular equation.

21. SOLUTION CONTINUED e. This is the graph of f, which agrees with the given graph. Note that the calculated slope, -2.6 is negative, which corresponds to the fact that f(x) decreases as x increases. Note that you could have solved the systems of equations in this example using matrices. The given system Write the system in matrix form a = 2.6 and b = 32 Multiply both sides by the inverse matrix. 19 = 5a + b 6 = 10a + b Complete the matrix multiplication

22. EXAMPLE 2 For the function graphed: a.Identify the kind of function it could be. b.On what interval or intervals is the function increasing or decreasing? Which way is the graph concave up or down? c. Describe something in the real world that a function with this shape graph could model. d.Find the particular equation of the function, given points (2, 89) and (3, 94) are on the graph. e.Confirm by plotting that your equation give the same graph.

23. SOLUTION a.The function could be quadratic because it has a vertex. b. The function is increasing for x 3, and it is concave down. c. The function could model anything that rises to a maximum and then falls back down again, such as the height of a ball as a function of time or the grade you could earn on a final exam as a function of how long you study for it. (Cramming too long might lower your score because of your being sleepy from staying up late.

24. SOLUTION CONTINUED d. y = ax + bx + c Write the general equation. 76 = a + b + c 89 = 4a + 2b + c 94 = 9a + 3b + c Substitute the given values of x and y into the equation of f. Solve by matrices y = 4x + 25x + 55 Write the equation. e. Plotting the graph confirms that the equation is correct. Note that the value of a is negative, which corresponds to the fact that the graph is concave down.

25. WARM UP d.Find a particular equation of the function if the given points are on the graph. e.Confirm by plotting that your equation gives the graph shown. The quadrant I part of a function graph is shown: a.Identify the type of function it could be. b.On what interval or intervals is the function increasing or decreasing, and which way is the graph concave? c.What relationship in the real world could be modeled by a function with this shape.

26. SOLUTION d.y = 2x – 7 a.Linear b.There is no concavity. The graph is increasing for all real values of x c.It could be the graph of a the height of a tree as it grows from a seed into a full grown tree.

27. EXAMPLE 3 For the function graphed: a.Identify the kind of function it could be. b.On what interval or intervals is the function increasing or decreasing? Which way is the graph concave up or down? c. Describe something in the real world that a function with this shape graph could model. d.Find the particular equation of the function, given points (4, 44.8) and (6, 151.2) are on the graph. e.Confirm by plotting that your equation give the same graph.

28. SOLUTION a.The function could be a power function or an exponential function. It is a power function because the graph appears to contain the origin, which exponential functions don’t do unless they are translated in the y-direction. b. The function is increasing over its entire domain shown and the graph is concave up everywhere. c. The function could model anything that starts at zero and increases at an increasing rate, such as the distance it takes a car to stop as a function of its speed when the driver applies the brakes, or the volumes of geometrically similar objects as a function of their length.

29. SOLUTION CONTINUED d. y = ax Write the untranslated general equation. 44.8 = a  4 151.2 = a  6 Substitute the given values of x and y into the equation. Divide the second equation by the first to eliminate a. 3.375 = 1.5 The a’s cancel, and 6/4 = 1.5 log 3.375 = log 1.5 Take the logarithm of both sides to get b out of the exponent. log 3.375 = b log 1.5

30. SOLUTION CONTINUED Substitute 3 for b in one of the equations. e. Plotting the graph confirms that the equation is correct. Note that the value of b is greater than 1, which corresponds to the fact that the graph is concave up. Write the particular equation. y = 0.7x 3

31. EXAMPLE 4 For the function graphed: a.Identify the kind of function it could be. b.On what interval or intervals is the function increasing or decreasing? Which way is the graph concave up or down? c. Describe something in the real world that a function with this shape graph could model. d.Find the particular equation of the function, given points (2, 10) and (5, 6) are on the graph. e.Confirm by plotting that your equation give the same graph.

32. SOLUTION a.The function could be exponential or quadratic, but exponential is picked because the graph appears to approach the x-axis asymptotically. b. The function is decreasing and concave up over its entire domain. c. The function could model any situation in which a variable quantity starts at some nonzero value and coasts downward, gradually approaching zero, such as the number of degrees a cup of coffee is above room temperature as a function of time since it started cooling.

33. SOLUTION CONTINUED d. y = ab Write the untranslated general equation. 10 = ab 6 = ab Substitute the given values of x and y into the equation. Divide the second equation by the first to eliminate a. 0.6 = b Raise both sides to the 1/3 power to eliminate the exponent of b. 0.6 = b Store without rounding b = 0.8434…

34. SOLUTION CONTINUED Substitute 0.8434…for b in one of the equations. e. Plotting the graph confirms that the equation is correct. Note that the value of b is between 0 and 1, which corresponds to the fact that the function is decreasing. Write the particular equation. a = 14.0572… Store without rounding y = 14.0572…(0.8434…) x

35. CH. 7.1/7.2 HOMEWORK Textbook pg. 293 #1-4 & pg. 301 #2-22 even Journal Questions: From what your have learned in this section, what do you consider to be the main idea? What is the difference between the parent quadratic function and any other quadratic function? How does the y-intercept of an exponential function differ from the y-intercept of a power function