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 \(f\):

  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:

  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.