6-1 Multiple Representations of Functions Warm Up Holt McDougal Algebra 2 Holt Algebra 2

I can: • Express a function using function notation. • Evaluate values in the domain of a function. • Interpret functions in function notation based on context.

An amusement park manager estimates daily profits by multiplying the number of tickets sold by 20. This verbal description is useful, but other representations of the function may be more useful. These different representations can help the manager set, compare, and predict prices.

Because each representation of a function (words, equation, table, or graph) describes the same relationship, you can often use any representation to generate the others.

A function is a special type of relation that pairs each domain value with exactly one range value.

The domain of a relation is the set of first coordinates (or x-values) of the ordered pairs. The range of a relation is the set of second coordinates (or y-values) of the ordered pairs. The domain of the track meet scoring system is {1, 2, 3, 4}. The range is {5, 3, 2, 1}.

Recreation Application Janet is rowing across an 80-meter-wide river at a rate of 3 meters per second. Create a table, an equation, and a graph of the distance that Janet has remaining before she reaches the other side. When will Janet reach the shore?

Continued Step 1 Create a table. Let t be the time in seconds and d be Janet’s distance, in meters, from reaching the shore. Janet begins at a distance of 80 meters, and the distance decreases by 3 meters each second.

Continued Step 2 Write an equation. is equal to minus Distance 80 3 meters per second. d = 80 – 3t

Step 3 Find the intercepts and graph the equation. Example 2 Continued Step 3 Find the intercepts and graph the equation. d-intercept:80 Solve for t when d = 0 d = 80 – 3t 0 = 80 – 3t t = = 26 –80 –3 2 3 t-intercept: 26 2 3 Janet will reach the shore after seconds. 26 2 3

Evaluating Functions: On my website https://www.tranlordylordylordmath.weebly.com Click on Evaluating Function Practice View the 3 videos examples 1-3 Then click on Understanding Function Notation for practice

Practice Interpretation and Evaluation in Real World Applications Click on Interpreting Domain and Range Worksheet

How do we know whether a given function is: 1. Linear 2. Quadratic 3. Exponential This is Math 1/Algebra 1 Review

Example 3: Using Multiple Representations to Solve Problems A hotel manager knows that the number of rooms that guests will rent depends on the price. The hotel’s revenue depends on both the price and the number of rooms rented. The table shows the hotel’s average nightly revenue based on room price. Use a graph and an equation to find the price that the manager should charge in order to maximize his revenue.

Example 3A Continued The data do not appear to be linear, so check finite differences.

Example 3A Continued Revenues 21,000 22,400, 23,400 24,000 First differences 1,400 1,000 600 Second differences 400 400 Because the second differences are constant, a quadratic model is appropriate. Use a graphing calculator to perform a quadratic regression on the data.

Example 3B: Using Multiple Representations to Solve Problems An investor buys a property for $100,000. Experts expect the property to increase in value by about 6% per year. Use a table, a graph and an equation to predict the number of years it will take for the property to be worth more than $150,000.

Example 3B Continued Make a table for the property. Because you are interested in the value of the property, make a graph, by using years t as the independent variable and value as the dependent variable.

The data appears to be exponential, so check for a common ratio. Example 3B Continued The data appears to be exponential, so check for a common ratio. r1 = = = 1.06 y1 y0 106,000 100,000 r2 = = = 1.06 y2 y1 112,360 106,000 r3 = = ≈ 1.06 y3 y2 119,102 112,360 r4 = = ≈ 1.06 y4 y3 126,248 119,102 Because there is a common ratio, an exponential model is appropriate. Use a graphing calculator to perform an exponential regression on the data.