Exponential Functions Quadratic Functions Linear Functions Review 8.5 Exponential Functions Quadratic Functions Linear Functions

Identifying from an equation: Linear Has an x with no exponent. y = 5x + 1 y = ½x 2x + 3y = 6 Quadratic Has an x2 in the equation. y = 2x2 + 3x – 5 y = x2 + 9 x2 + 4y = 7 Exponential Has an x as the exponent. y = 3x + 1 y = 52x 4x + y = 13

Examples: LINEAR, QUADRATIC or EXPONENTIAL? y = 6x + 3 y = 7x2 +5x – 2

Examples: LINEAR, QUADRATIC or EXPONENTIAL? y = 6x + 3 EXPONENTIAL y = 7x2 +5x – 2 QUADRATIC 9x + 3 = y LINEAR 42x = 8 EXPONENTIAL

Identifying from a graph: Linear Makes a straight line Quadratic Makes a parabola Exponential Rises or falls quickly in one direction

LINEAR, QUADRATIC or EXPONENTIAL? a) b) c) d)

LINEAR, QUADRATIC or EXPONENTIAL? a) quadratic b) exponential c) linear d) absolute value

absolute value Think of the definition of absolute value. It is a piecewise defined function. |x| = x, if x>=0 and -x if x<0. In other words, the graph of y=|x| is formed by two pieces of two lines. For the part of the domain where x-values are less than zero, the graph corresponds to the graph of y=-x. For parts of the domain where x-values are greater than or equal to zero, the graph corresponds to the graph of y=x. While the absolute value function does not satisfy the definition for a linear function, it is actually "parts" of two linear functions.

Is the table linear, quadratic or exponential? y changes more quickly than x. Never see the same y value twice. Common multiplication pattern Linear Never see the same y value twice. 1st difference is the same Quadratic See same y more than once. 2nd difference is the same

Concept

Example 1 A. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (1, 2), (2, 5), (3, 6), (4, 5), (5, 2) Answer: The ordered pairs appear to represent a quadratic equation.

Example 1 B. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–1, 6), (0, 2), Answer: The ordered pairs appear to represent an exponential function.

Example 1 A. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–2, –6), (0, –3), (2, 0), (4, 3) A. linear B. quadratic C. exponential

Example 1 B. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. (–2, 0), (–1, –3), (0, –4), (1, –3), (2, 0) A. linear B. quadratic C. exponential

Example 2 A. Look for a pattern in the table of values to determine which kind of model best describes the data. –1 1 3 5 7 2 2 2 2 First differences: Answer: Since the first differences are all equal, the table of values represents a linear function.

Example 2 B. Look for a pattern in the table of values to determine which kind of model best describes the data. 36 12 4 __ 4 3 9 –24 –8 –2 __ 2 3 __ 8 9 – First differences: The first differences are not all equal. So, the table of values does not represent a linear function. Find the second differences and compare.

Example 2 –24 –8 –2 2 3 8 9 – First differences: 16 5 1 3 1 7 9 __ 2 3 8 9 – First differences: 16 5 __ 1 3 1 __ 7 9 Second differences: The second differences are not all equal. So, the table of values does not represent a quadratic function. Find the ratios of the y-values and compare. 36 4 __ 9 12 3 __ 1 3 __ 1 3 __ 1 3 __ 1 3 Ratios:

Example 2 The ratios of successive y-values are equal. Answer: The table of values can be modeled by an exponential function.

Example 2 A. Look for a pattern in the table of values to determine which kind of model best describes the data. A. linear B. quadratic C. exponential D. none of the above 17 33 57 89 8 16 24 32 8 8 8

Example 2 B. Look for a pattern in the table of values to determine which kind of model best describes the data. A. linear B. quadratic C. exponential D. none of the above

Example 2 B. Look for a pattern in the table of values to determine which kind of model best describes the data. A. linear B. quadratic C. exponential D. none of the above 24 6 3/2 3/8 4 4 4 4

Example 3 Write an Equation Determine which kind of model best describes the data. Then write an equation for the function that models the data. Step 1 Determine which model fits the data. –1 –8 –64 –512 –4096 –7 –56 –448 –3584 First differences:

Example 3 –7 –56 –448 –3584 First differences: –49 –392 –3136 Write an Equation –7 –56 –448 –3584 First differences: –49 –392 –3136 Second differences: –1 –8 –64 Ratios: –512 –4096 × 8 × 8 × 8 × 8 The table of values can be modeled by an exponential function.

Example 3 Write an Equation Step 2 Write an equation for the function that models the data. The equation has the form y = abx. Find the value of a by choosing one of the ordered pairs from the table of values. Let’s use (1, –8). y = abx Equation for exponential function –8 = a(8)1 x = 1, y = –8, b = 8 –8 = a(8) Simplify. –1 = a An equation that models the data is y = –(8)x. Answer: y = –(8)x

Example 3 Determine which model best describes the data. Then write an equation for the function that models the data. 0 3 12 27 48 3 9 15 21 6 6 6 A. quadratic; y = 3x2 B. linear; y = 6x C. exponential; y = 3x D. linear; y = 3x

Write an Equation for a Real-World Situation Example 4 KARATE The table shows the number of children enrolled in a beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data.

Write an Equation for a Real-World Situation Example 4 Understand We need to find a model for the data, and then write a function. Plan Find a pattern using successive differences or ratios. Then use the general form of the equation to write a function. Solve The first differences are all 3. A linear function of the form y = mx + b models the data.

Example 4 y = mx + b Equation for linear function Write an Equation for a Real-World Situation Example 4 y = mx + b Equation for linear function 8 = 3(0) + b x = 0, y = 8, and m = 3 b = 8 Simplify. Answer: The equation that models the data is y = 3x + 8. Check You used (0, 8) to write the function. Verify that every other ordered pair satisfies the function.

Example 4 WILDLIFE The table shows the growth of prairie dogs in a colony over the years. Determine which model best represents the data. Then write a function that models the data. A. linear; y = 4x + 4 B. quadratic; y = 8x2 C. exponential; y = 2 ● 4x D. exponential; y = 4 ● 2x

EXAMPLE 2 Identify functions using differences or ratios b. x – 2 – 1 1 2 y 4 7 10 Differences: 3 3 3 3 ANSWER The table of values represents a linear function.

EXAMPLE 2 Identify functions using differences or ratios Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. a. x –2 –1 1 2 y –6 –4 6 First differences: 0 2 4 6 Second differences: 2 2 2 ANSWER The table of values represents a quadratic function.

GUIDED PRACTICE for Examples 1 and 2 2. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. y 2 x – 2 – 1 1 0.08 0.4 10 ANSWER exponential function

Is the table linear, quadratic or exponential? y 1 2 -1 3 4 5 8 x y 1 3 2 9 27 4 81 5 243 x y 1 5 2 9 3 13 4 17 21 Linear. quadratic exponential