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Quizzes > High School Quizzes > Electives

Unit 2 Functions and Graphs Practice Quiz

Master graphing skills with detailed quiz answers

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting Graph Genius Unit 2, a math trivia for high school students.

What is the domain of a function?
The set of all output values
A description of a function's behavior
The set of intersection points on a graph
The set of all possible input values
The domain of a function consists of all the valid input values for which the function is defined. This is a fundamental concept when analyzing functions and their graphs.
Identify the y-intercept of the line y = 2x + 3.
(0, 3)
None of the above
(3, 2)
(2, 3)
The y-intercept is the point where the graph meets the y-axis, which occurs when x is 0. Substituting x = 0 into the equation gives y = 3.
What shape best describes the graph of y = x²?
A circle
A parabola opening upward
A parabola opening downward
A straight line
The graph of y = x² is a classic U-shaped parabola that opens upward due to the positive coefficient of the squared term. This is a basic example used to introduce quadratic functions.
What does a negative slope indicate about a linear graph?
The line is horizontal
The line has a steep curve
The line is decreasing, falling as it moves from left to right
The line is increasing, rising as it moves from left to right
A negative slope means that as the x-values increase, the y-values decrease, resulting in a line that falls from left to right. This is a key characteristic of linear functions with negative rates of change.
Which of the following describes an increasing function?
The function always returns a constant value
For any two inputs x₝ < x₂, f(x₝) > f(x₂)
For any two inputs x₝ < x₂, the function satisfies f(x₝) < f(x₂)
The graph has a negative slope
An increasing function is one where a larger input always leads to a larger output. This means if x₝ is less than x₂, then f(x₝) is less than f(x₂), ensuring the function steadily rises.
Identify the vertex of the quadratic function f(x) = x² - 4x + 1.
(2, -3)
(-2, 3)
(1, -4)
(4, 1)
The vertex of a quadratic function in the form f(x) = ax² + bx + c is found using the formula x = -b/(2a). For this function, x = 4/2 = 2, and substituting back gives y = -3, so the vertex is (2, -3).
What transformation is represented by the function g(x) = (x - 3)² + 2 relative to f(x) = x²?
The graph shifts 3 units to the left and 2 units downward
The graph shifts 3 units to the left and 2 units upward
The graph shifts 3 units to the right and 2 units downward
The graph shifts 3 units to the right and 2 units upward
In the function g(x) = (x - 3)² + 2, the expression (x - 3) indicates a horizontal shift 3 units to the right, while the +2 indicates a vertical shift upward by 2 units. These transformations change the position of the graph without altering its shape.
Evaluate f(x) = 3(x + 1)² - 5 at x = -1.
5
-5
0
3
Substitute x = -1 into the function: 3(0)² - 5 equals -5. The squared term becomes zero, leaving only the constant -5.
What is the effect of multiplying an entire function by -1 on its graph?
It reflects the graph over the x-axis
It translates the graph upward
It reflects the graph over the y-axis
It compresses the graph horizontally
Multiplying a function by -1 negates all the output values, resulting in a reflection over the x-axis. This transformation inverts the graph vertically while keeping the x-values unchanged.
Which of the following best describes the inverse of the function f(x) = 2x + 1?
A line with slope -2 and y-intercept -1
A line with slope 2 and y-intercept 1
A line with slope -1/2 and y-intercept 1/2
A line with slope 1/2 and y-intercept -1/2
To find the inverse of f(x) = 2x + 1, switch x and y and solve: x = 2y + 1, so y = (x - 1)/2. The inverse is a line with a slope of 1/2 and a y-intercept of -1/2, and it is also the reflection of f(x) over the line y = x.
For the absolute value function f(x) = |x - 2|, what is the value of f(5)?
7
2
-3
3
Substitute x = 5 into the equation: |5 - 2| equals |3|, which is 3. The absolute value function returns the non-negative distance between the number and 2.
Which function represents a vertical stretch of y = sin(x) by a factor of 3?
y = 3 sin(x)
y = sin(x) + 3
y = sin(x)/3
y = sin(3x)
Multiplying the sine function by 3 increases its amplitude by a factor of three while leaving the period unchanged. This vertical stretch makes the peaks and valleys three times as high and deep.
What transformation moves the graph of y = x³ to obtain the graph of y = (x - 4)³?
A vertical shift 4 units downward
A vertical shift 4 units upward
A horizontal shift 4 units to the right
A horizontal shift 4 units to the left
Replacing x with (x - 4) in the cubic function translates the graph horizontally 4 units to the right. The shape of the graph remains unchanged except for the shift.
A function has a domain of all real numbers and a range of all non-negative numbers. Which type of function could it be?
An exponential decay function
A quadratic function in vertex form that opens upward
A linear function with a negative slope
An absolute value function with a reflection
Quadratic functions that open upward typically have a minimum point, which makes their range all non-negative numbers provided the vertex is at or above zero. The domain of quadratics is all real numbers.
What is the domain of the function f(x) = 1/(x² - 9)?
All non-zero real numbers
All real numbers except x = 3 and x = -3
All positive real numbers
All real numbers except x = 9
The function f(x) = 1/(x² - 9) is undefined when the denominator is zero. Since x² - 9 factors to (x - 3)(x + 3), x cannot be 3 or -3.
Find the x-intercepts of the quadratic function f(x) = x² - 5x + 6.
x = 2 and x = 3
x = 0 and x = 6
x = -2 and x = -3
x = 1 and x = 6
Factoring f(x) = x² - 5x + 6 gives (x - 2)(x - 3) = 0, which means x = 2 or x = 3. These are the points where the graph intersects the x-axis.
Determine the vertical asymptote of the function f(x) = (2x - 4)/(x - 3).
x = 2
x = 4
x = 3
x = -3
A vertical asymptote occurs where the denominator of a function equals zero (and the numerator is non-zero). Since x - 3 = 0 when x = 3, the vertical asymptote is at x = 3.
What is the domain of the function f(x) = √(x + 2)?
x ≤ -2
x > -2
All real numbers
x ≥ -2
The square root function is defined only when its argument is non-negative. For f(x) = √(x + 2), the condition x + 2 ≥ 0 must hold, meaning x must be greater than or equal to -2.
Which transformation is applied to y = x² to obtain f(x) = -(x + 2)² + 3?
Reflection over the x-axis, 2 units left, and 3 units up
Horizontal compression and 3 units up
Vertical stretch by -1, 2 units right, and 3 units up
Reflection over the y-axis, 2 units right, and 3 units down
The function f(x) = -(x + 2)² + 3 applies a reflection over the x-axis due to the negative sign. The (x + 2) indicates a horizontal shift 2 units to the left, and the +3 represents a vertical shift upward by 3 units.
Determine the vertex of the quadratic function y = -2x² + 8x - 3.
(-2, -5)
(2, -5)
(2, 5)
(-2, 5)
The vertex of a quadratic function can be found using the formula x = -b/(2a). For y = -2x² + 8x - 3, x = -8/(-4) = 2, and substituting x = 2 gives y = 5. Thus, the vertex is at (2, 5).
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Study Outcomes

  1. Analyze the key features of function graphs, including intercepts, slopes, and asymptotes.
  2. Interpret the behavior of functions by examining their graphical representations.
  3. Apply transformations to basic functions and predict resulting changes in their graphs.
  4. Synthesize graphical information to solve problems involving domain and range.
  5. Evaluate the impact of shifts and stretches on the form and position of function graphs.

Unit 2 Functions & Graphs Cheat Sheet

  1. Understand the definition of a function - A function pairs every input with exactly one output, so you'll never be left guessing which value goes where. Think of it like a vending machine that always drops just one snack when you push a button. This clarity is what sets functions apart from general relations. Lumen Learning summary
  2. Master function notation - The notation f(x) reads as "f of x" and tells you the output associated with a specific input x. For instance, if f(x)=2x+3, plugging in x=2 yields f(2)=7, like a math recipe producing tasty results. Getting comfortable with this shorthand helps you communicate ideas precisely. Pearson Precalculus guide
  3. Learn to evaluate functions - Evaluating a function means substituting a given x-value into f(x) to find the corresponding output y. It's like plugging ingredients into a formula to bake a cake, where the final product depends on what you put in. Mastering this skill is essential for sketching graphs and solving equations. Lumen Learning summary
  4. Familiarize yourself with common function types - Linear, quadratic, and cubic functions each have signature shapes - a line, a parabola, and an S-curve respectively - that help you recognize their behavior at a glance. Spotting these patterns is like identifying different dance moves on the math stage. The more types you know, the faster you can interpret graphs. BYJU'S Functions & Graphs
  5. Apply the vertical line test - If you can draw a vertical line that crosses a graph more than once, it fails the function test. It's like a "no double-dips" rule - only one bite per stick. Using this quick check ensures you only work with true functions. Symbolab Function Characteristics
  6. Use the horizontal line test - To check if a function is one-to-one, slide a horizontal line across its graph; more than one intersection means it's not invertible. Think of it like a speed camera: it can only lock onto one car at a time. One-to-one functions guarantee a unique inverse. Symbolab One-to-One Test
  7. Understand domain and range - The domain lists all allowable inputs (x-values), while the range covers possible outputs (y-values). It's like the guest list and gift table of a math party - each plays a role in defining the event's boundaries. Knowing these sets keeps your functions well-defined. Lumen Learning summary
  8. Recognize transformations of functions - Shifts, reflections, stretches, and compressions let you tweak a parent graph to create new shapes, like tuning an instrument to hit different notes. For example, f(x)+k moves the graph up by k units, making your math masterpiece uniquely yours. Lumen Learning cheat sheet
  9. Practice identifying intercepts - X-intercepts occur where the graph crosses the x-axis (f(x)=0), and y-intercepts show up at x=0. They act as anchoring points, much like mooring buoys for your curve. Finding these makes plotting functions a breeze. Lumen Learning summary
  10. Explore inverse functions - Inverses undo each other's effects: if f(g(x))=x and g(f(x))=x, you've got a pair of perfect mathematical dance partners. Sketching inverses also means reflecting the graph across the line y=x. Understanding this concept unlocks powerful solution strategies. Lumen Learning cheat sheet
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