Study Guide for Exam 2

This will be a closed-book exam. You may use a calculator, but a calculator is not required.

To do well on the exam, you should be able to do the following:

Section 2.1: Functions

1. List examples of functions from everyday life and biology.

2. Specify what makes a function more specific than just “a rule that assigns outputs to inputs”.

3. Give the definition of a function.

4. Explain how the vertical line test can be used to verify if a curve in the coordinate plane is the graph of a function, and relate this to the definition of a function.

5. Define the domain and range of a function.

6. Given a function $f$:

   • Express in words how $f$ acts on an input $x$ to generate an output $f(x)$.
   • Evaluate the function at a specified input.
   • Find the domain and range of $f$.

7. Use the ‘box’ notation on page 150 of the text as a way to think about a variable as a ‘blank space’ for the input to a function.

8. Demonstrate the correspondence between verbal, algebraic, visual, and numerical representations of a function by converting from one representation to another.

9. Evaluate a piecewise function by reading left-to-right: first determine the condition on the input $x$, then determine the output under that condition.

Section 2.4: Average Rate of Change of a Function

1. Define the net change of a function from $x = a$ to $x = b$.
2. Define the average rate of change of a function $f$ between $x = a$ and $x = b$ in terms of $a, b, f(a)$, and $f(b)$.
3. Define the average rate of change of a function as the ‘rise’ of the function over the ‘run’ of the function.
4. Relate the average rate of change of a function $f$ to the slope of the line segment between $(a, f(a))$ and $(b, f(b))$.

Section 2.5: Linear Functions and Models

1. Explain the meaning of the statement: “the only things linear functions can do is go up, go down, or stay flat.”

2. Specify the standard form of a linear function.

3. Given a linear function in non-standard form, rewrite the function in standard form.

4. Given a function, determine whether the function is linear or nonlinear.

5. Graph a linear function given one of:

   • The equation for the function
   • The slope and intercept of the function
   • The slope and a point on the graph of the function
   • Two points on the graph of the function

6. Compute the slope of a linear function using the average rate of change of the function.

7. Compute the slope of a linear function as ‘rise over run.’

8. Determine the form of a linear function from a ‘word problem’ given the requisite information to determine the equation of a line.

Section 3.1: Quadratic Functions and Models

1. Identify whether a quadratic function opens up or down using its coefficients.
2. Identify where a quadratic function obtains a particular value.
3. Express a quadratic function given as $f(x) = a x^{2} + b x + c$ in standard form $f(x) = a(x - h)^2 + k$, using the result from the top of page 249 of Stewart, Redlin, and Watson.
4. Define the properties of a global minimum / maximum of a function.
5. Identify when a quadratic function has a global maximum / minimum, and identify where the maximum / minimum occurs and its value.
6. Identify when a quadratic function is an appropriate model for a scientific question, locate the relevant information needed to construct the quadratic function, and identify the relevant characteristics of the quadratic function for answering the original question.

Section 3.2: Polynomial Functions and Their Graphs

1. Determine the degree, coefficients, and leading term of a given polynomial function.
2. Given a graph of a function, determine whether the function could be a polynomial function.
3. Determine the end behavior of a given polynomial function.
4. Given a factored polynomial function $P$, determine the roots / zeros / horizontal intercepts of the polynomial function.
5. Given the graph of a polynomial function $P$, determine the roots / zeros / horizontal intercepts of the polynomial function.
6. Sketch the graph of a given polynomial function using the method in the grey box at the bottom of page 259 of Stewart, Redlin, and Watson.