MATH11

• The first multiple choice question is: For the function f(x) = 3x^2 - 4x + 1, what is the value of f(-1)?
• f(x)


  Function notation, represents the value of y when x is a certain value
• Evaluating f(-1)


  1. Replace x with -1 in the original function

  2. Evaluate the expression 3(-1)^2 - 4(-1) + 1

  3. Simplify to get the answer 8
• The answer to question 1 is C. 8
• Question 2 is about finding the range of the function f(x) = -5x^2 + 4
• Range of a function


  • The set of all possible y-values the function can take
• Finding the range of f(x) = -5x^2 + 4


  1. Sketch the graph of the parabola

  2. Observe that the vertex is at (0,4)

  3. The y-values will be less than or equal to 4

  4. Therefore, the range is y ≤ 4
• The answer to question 2 is C. y ≤ 4
• Question 3 is about determining which of the given relations is not a function
• Function


  A relation where each x-value has only one corresponding y-value
• Checking if a relation is a function


  1. Use the vertical line test

  2. If a vertical line intersects the relation in more than one point, it is not a function
• Part B of the question fails the vertical line test, so it is not a function
• The answer to question 3 is B. Part B
• Question 4 is about determining how many points the curve y = 3x^2 - 4x + 8 and the line y = 3x + 5 intersect
• Solving for the intersection points


  1. Set the two equations equal to each other

  2. Solve the resulting quadratic equation

  3. The number of real solutions determines the number of intersection points
• The discriminant B^2 - 4AC is greater than 0, so there are two intersection points
• The answer to question 4 is C. 2 points
• Question 5 is about finding the value of a for the function f(x) = 2x^2 - 5x + 1 if f(a) = 4
• Solving for a when f(a) = 4

  1. Substitute 4 for f(a) in the original function

  2. Solve the resulting quadratic equation for a

  3. The solution is a = 3
• The answer to question 5 is B. 3
• Question 6 is about finding the restrictions on the rational expression (2x^2 + 5x + 3) / (4x^2 - 9)
• Restrictions on a function


  The values of x that make the function undefined
• Finding the restrictions


  1. Set the denominator equal to 0 and solve for x

  2. The solutions are x = ±3/2, which are the restrictions
• The answer to question 6 is C. x = ±3/2
• Question 7 is about the horizontal translation of the graph of g(x) = 3f(2x - 2) - 1 based on the graph of f(x)
• Horizontal translation


  Shifting the graph left or right
• Identifying the horizontal translation


  1. Factor out the coefficient of x, which is 2

  2. The remaining expression is x - 1, so the horizontal translation is 1 unit to the right
• The answer to question 7 is D. 1 unit to the right
• Question 8 is about finding the coterminal angle for 240 degrees
• Coterminal angles


  Angles that have the same terminal arm
• Finding the coterminal angle

  1. Draw the angle of 240 degrees

  2. Subtract or add 360 degrees to get an angle with the same terminal arm

  3. The coterminal angle is -120 degrees
• The answer to question 8 is B. -120 degrees
• Question 9 is about finding the related acute angle for 210 degrees
• Related acute angle


  The angle between the terminal arm and the closest x-axis
• Finding the related acute angle

  1. Draw the angle of 210 degrees

  2. The angle between 180 degrees and 210 degrees is 30 degrees
• The answer to question 9 is A. 30 degrees
• Question 10 is about the relationship between the graphs of sine and cosine functions
• Cosine function


  The same as a sine function translated 90 degrees to the left
• The answer to question 10 is A. The graph of the cosine function is the same as the graph of a sine function that has been translated 90 degrees to the left
• Question 11 is about finding the maximum value of the function y = sin(x) - 6