1.1 Change in tandem
For each of the following relations:
Find the domain.
Find the range.
Determine whether or not the relation is a function.
For each of the following relations:
Find the domain.
Find the range.
Determine whether or not the relation is a function.
| x | 1 | 3 | 5 | 6 | 2 |
|---|---|---|---|---|---|
| y | 3 | 2 | 7 | 1 | 2 |
| x | 4 | 4 | 8 | 5 | 3 |
|---|---|---|---|---|---|
| y | 1 | 9 | 3 | 2 | 6 |
| x | 2 | 6 | 8 | 1 | 3 |
|---|---|---|---|---|---|
| y | 6 | 5 | 1 | 8 | 3 |
| x | 7 | 7 | 7 | 7 | 7 |
|---|---|---|---|---|---|
| y | -2 | 2 | 3 | 5 | 7 |
Consider the graph of y = x+ 5:
State the domain.
State the range.
Consider the graph of y = \dfrac{2x}{3}:
State the domain.
State the range.
Given the function f\left(x\right) = x^2 + 4x - 6.
Create a table of values for x and f(x) where x = -2,\, -1,\, 0,\, 1,\, and 2.
Sketch a graph of the function.
Analyze how the output values change as the input values increase.
Consider the function f(x) = x^{3} - 3x^{2} + 2x.
Sketch a rough graph of f(x).
Determine the intervals over which the graph of f(x) is concave up or down.
Draw a tangent line at x=2 to show the rate of change at this point.
Explain the significance of the rate of change of f(x) at x = 2.
Consider the function f(x) = x^{2} - 4{x} - 5.
Identify the zeros of the function f(x) on the graph.
Explain what the zeros represent in terms of the function.
Sketch a rough graph of f(x), clearly indicating the zeros.
Consider the following graph of f\left(x\right):
Is f\left(x\right) increasing, decreasing, or constant at x=6. Explain.
Is the rate of change of f\left(x\right) increasing, decreasing, or constant at x=6. Explain.
State the interval where f\left(x\right) is concave up.
State the interval where f\left(x\right) is concave down.
Consider the function that represents the amount of money John has saved over time, given by S(t) = 100t + 500, where S is the amount saved in dollars and t is time in months.
Interpret the function in the context of this scenario.
Identify the domain and range of the function in this context.
Identify the independent and dependent variables of the function and explain why they are classified as such.
Suppose John has been saving for 10 months. Using the function, determine how much money John has saved.
A bakery sells cupcakes and the price per cupcake depends on the number of cupcakes bought. This relationship can be represented by the function P(n) = - 0.1n^{2} + 2n, where P is the price in dollars and n is the number of cupcakes bought.
Suppose you buy 10 cupcakes. Using the function rule, find out how much you would have to pay.
Looking at the function, is the cost for each additional cupcake you buy constant? Explain your reasoning.
Suppose you paid \$9 at the bakery. Using the function rule, find out approximately how many cupcakes you bought?
A car starts from rest and accelerates uniformly. After 5 seconds, it has reached a speed of 20 meters per second. It continues to accelerate at the same rate for another 10 seconds.
Construct a graph that represents the described changes. Use the x-axis to represent time (t) in seconds and the y-axis to represent speed (v) in meters per second.
Label the axes indicating the appropriate scales.
Plot the data points to show the initial acceleration and the subsequent constant acceleration.
Connect the data points smoothly to represent the change in speed over time.
The number of fish in a certain lake over time can be modeled by the function F(t) = 15t^2 -200t +5000, where F is the number of fish and t is the time in years since observations began.
Use the function rule to find the fish population after t = 2 years and t = 6 years.
Is the function increasing or decreasing over the interval from t = 2 years to t = 6 years? If so, interpret the increase or decrease in this situation.
Identify any time intervals where the fish population is increasing. Justify your reasoning based on the function rule.
Interpret the increase in this situation.
Consider the function f(x) = 2x^{3} - 3x^{2} + 2x - 1.
Plot the graph of the function f(x) on a set of coordinate axes.
Draw a tangent line to points x=-1,0, and x=1.
Interpret the relationship between the graph of the function and its rate of change.
Discuss any patterns or trends you observe in the rate of change as you move along the graph of the function.
Consider the following graph of a function f \left( x \right):
Determine the domain, range, independent variable, and dependent variable of the function.
Identify the output when the input is x = 1 and the input when the output is y = 0.
Describe the intervals where the function is increasing and decreasing.
Researchers have developed a new function to estimate the maximum heart rate, H(x), in \text{ bpm}, for women based on their age, x, in years. The function is given by H(x) = - 0.92x + 208.
Use the function H(x) to find the estimated maximum heart rate for a woman who is 32 years old.
Interpret your answer in the context of this problem. What does the estimated maximum heart rate of H(32) represent for a 32-year-old woman?
A ball is thrown straight up into the air. Its height h(t) in meters above the ground is modeled by the function h(t) = -5t^2 + 25t, where t is the time in seconds.
Construct a graph of the function using input-output pairs.
Determine the approximate maximum height reached by the ball and the time it takes to reach that height.
Identify the zeros of the function and explain what they represent in terms of the ball's height and time.
The plotted graph provides data about the relationship between the number of hours studied per week (hours) and the grade point average (GPA) of students.
The graph shows an exponential growth where the function can be modeled as f(x)=a^x. Identify the approximate function rule using one of given coordinates.
Describe the relationship between the number of hours studied per week and the GPA of students using the function rule.
The temperature of a cup of coffee is modeled by the function T(t) = - 50e^{-0.1t} + 70, where T(t) is the temperature (in degrees Celsius) at time t minutes after being poured.
Determine the domain, range, independent variable, and dependent variable of the function.
Calculate the temperature of the coffee after 5 minutes.
Explain what the zero of the function represents in terms of temperature and time.
A company's profit, P(x), in thousands of dollars, can be modeled by the function \\P(x) = -2x^3 + 12x^2 - 20x + 10, where x is the cost of their product in hundred dollars.
Create a graph of the function using x=0,\,1.1,\,...\,,4.
Determine the intervals where the profit is increasing and decreasing.
State the point of inflection.
State the intervals where f(x) is concave up and down.
Interpret the concavity of the function in terms of the company's profit and cost of their product.
State and interpret the zeros of f(x).
Explain the relationship between the graph of the function and its rate of change at x=4.
A population of bacteria is modeled by the function B(t) = 200e^{0.03t}, where B(t) is the number of bacteria at time t hours.
Determine the output for an input of t = 5 hours. Round up to the nearest whole number.
Construct a graph representing the population growth of the bacteria.
Describe the intervals over which the population is increasing and provide an explanation in terms of rate of change at x=3.