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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Which Graph Represents a Function?

Sharpen Your Skills: Identify Valid Function Graphs

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Colorful paper art promoting a dynamic math quiz, Graph Function Frenzy, for high school students.

Which of the following correctly describes the vertical line test?
A graph represents a function if it passes through the origin.
A graph represents a function if no vertical line intersects it more than once.
A graph represents a function if all horizontal lines intersect it at most once.
A graph represents a function if each input has at least two outputs.
The vertical line test states that a graph represents a function if every vertical line intersects the graph at most once. This ensures that each input corresponds to one unique output.
Given a graph with a continuous curve and only one y-value for each x-value, which statement is true?
It is a function only if it passes through the y-axis.
It is not a function if it is linear.
It is not a function because it is continuous.
It is a function.
A function is defined by having exactly one output for each input. Since each x in the graph gives one y-value, it qualifies as a function regardless of continuity.
Which feature invalidates a graph as representing a function?
A vertical line cutting through the graph twice.
A point where the graph touches the x-axis.
A sharp corner on the graph.
A horizontal line cutting through the graph twice.
The vertical line test reveals a relation is not a function if any vertical line crosses the graph more than once. This indicates one input is linked to multiple outputs.
Which of the following graphs would not represent a function?
A circle.
A straight line.
A sine wave.
A parabola opening upward.
A circle fails the vertical line test because some vertical lines will intersect it at two distinct points. Therefore, it does not meet the criteria for a function.
What does the vertical line test determine for a graph?
Whether the graph intersects the x-axis.
Whether the function has an inverse.
Whether the relation is a function.
Whether the function is increasing or decreasing.
The vertical line test is used to decide if a graph represents a function by ensuring every vertical line intersects the graph at most once. This confirms the uniqueness of outputs for each input.
When graphing the function f(x) = x², what is the shape of the graph?
Parabola opening upward.
Straight line.
Parabola opening downward.
Circle.
The function f(x) = x² produces a parabolic shape that opens upward. This is the standard graph of a quadratic function.
Consider a piecewise function defined by f(x) = x + 1 for x < 0 and f(x) = x² for x ≥ 0. Does its graph represent a function?
Yes, because piecewise graphs are always functions.
No, because it has a gap.
No, because it has two different expressions.
Yes, because each x-value corresponds to one y-value.
Even though the function is defined by different expressions, as long as each x-value produces a single, unique y-value, it qualifies as a function. Piecewise definitions do not hinder the function property as long as the domains do not overlap.
Which graph represents the inverse of a function that passes the vertical line test?
The reflection of the original graph across the y-axis.
The original graph shifted upwards.
The original graph without any changes.
The reflection of the original graph across the line y = x that also passes the vertical line test.
An inverse function is created by reflecting the original function across the line y = x. For the inverse to be a function, the reflected graph must also pass the vertical line test.
A graph shows a break at x = 2 but is otherwise continuous; is it still considered a function?
No, because any gap invalidates a function.
No, because a function must be continuous.
Yes, because discontinuities do not affect the function's definition.
Yes, only if the break is a removable discontinuity.
A function is defined by every input having exactly one output, not by continuity. Even if there is a break or jump, as long as each x-value has one unique y-value, the relation is a function.
For the function f(x) = √x, what is the domain based on its graph?
x > 0
x < 0
All real numbers
x ≥ 0
The square root function is only defined for non-negative values of x. This means the graph starts at x = 0 and includes all values greater than or equal to zero.
If a graph passes both the vertical and horizontal line tests, what property does the function have?
It is one-to-one and has an inverse function.
It is a quadratic function.
It is periodic.
It is only increasing.
A graph that passes the vertical line test is a function, and passing the horizontal line test indicates it is one-to-one. One-to-one functions have inverses that are also functions.
Which graph represents a constant function?
A diagonal line.
An upward curving line.
A vertical line.
A horizontal line.
A constant function always gives the same output, regardless of the input. Graphically, this is depicted as a horizontal line.
When a function's graph is reflected over the x-axis, when does it represent the original function?
Only if the function is symmetric about the y-axis.
Only if the function is even.
Only if the function is the zero function.
Yes, for any function.
Reflecting a graph over the x-axis generally changes the output values of the function. The only exception is the zero function, which remains unchanged after reflection.
Consider f(x) = 2 for rational x and f(x) = 3 for irrational x. Does the graph represent a function?
Yes, because it is continuous.
No, because some x-values would have two outputs.
No, because it uses two different formulas.
Yes, because each x-value produces a unique output.
For any specific x-value, it is either rational or irrational, so only one of the definitions applies. This ensures that every x-value is paired with exactly one y-value, fulfilling the requirement of a function.
Which transformation results in a vertical stretch of a function's graph?
Subtracting a constant from the function.
Multiplying the function by a constant greater than 1.
Adding a constant to the function.
Reflecting the function across the y-axis.
Multiplying a function by a constant greater than 1 increases the y-values, thus stretching the graph vertically. This transformation changes the graph's scale without affecting its horizontal position.
If a function passes both the vertical and horizontal line tests, what additional property does it have?
It is periodic.
It is symmetric about the y-axis.
It is discontinuous.
It is bijective and has an inverse that is also a function.
A function that passes the vertical line test is indeed a function, and if it also passes the horizontal line test, it is one-to-one (bijective). This guarantees the existence of an inverse function.
How does the horizontal line test relate to a function's invertibility?
It determines if a function has an inverse that is also a function.
It measures the function's rate of change.
It identifies whether the function is increasing.
It defines the domain of the function.
The horizontal line test checks if each output value is produced by only one input. If so, the function is one-to-one, ensuring that its inverse will also be a function.
A function f is graphed with a line segment from (-2, 3) to (0, 3) and another segment from (0, 4) to (2, 4). Is f a function, and why?
No, because the graph consists of two separate segments.
Yes, because every x-value has exactly one y-value.
Yes, because horizontal lines intersect both segments.
No, because the jump discontinuity means it is not a function.
Despite having two separate segments, the graph assigns only one y-value to each x-value in its domain. A jump or discontinuity does not disqualify a graph from representing a function.
If a vertical line intersects a graph at a point where one x-value corresponds to two different y-values, what is the implication?
It is a function if the two y-values are equal.
It indicates the graph is symmetric about the origin.
It fails the vertical line test and is not a function.
It still passes as a function if it looks continuous elsewhere.
A core property of functions is that each x-value maps to only one y-value. If any vertical line intersects the graph at two points with different y-values, the relation fails to be a function.
For a function represented by both a quadratic curve and a restricted line segment, how do you determine the domain?
By identifying all x-values that produce a unique y-value on either part of the graph.
By reflecting the graph over the y-axis.
By considering only the quadratic portion.
By taking the union of all y-values from both parts.
The domain of a function is the set of all x-values for which the function is defined with a unique output. This involves considering both segments of the graph and ensuring that each x-value only appears once.
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Study Outcomes

  1. Analyze graphs using the vertical line test to determine if they represent functions.
  2. Identify key characteristics that distinguish functions from non-functions.
  3. Apply reasoning to select the correct graph representing a function.
  4. Evaluate graphical information to interpret domain and range for given functions.

Function Graph Cheat Sheet

  1. Vertical Line Test - Think of a vertical line as an x-value spotlight: if it illuminates more than one point on the graph, you lose function status! This quick stunt guarantees each input only leads to one output. Try sketching curves and slicing them up to see who passes the test. Vertical Line Test Guide
    2. MathWorld: Vertical Line Test
  2. Linear Functions - Straight lines are the cool kids of the graph world and always ace the Vertical Line Test - no surprise there! Every x maps to exactly one y, making these functions smooth operators. Spotting them is as easy as checking for constant slope in their equation. Linear Functions Rundown
    3. Socratic: Vertical Line Test
  3. Circles - Circles can be sneaky: a vertical line can slice them at two points, which means they're not functions in the x→y sense. If you draw x²+y²=r², you'll see double y-values for many x's. Remember to watch out for top-and-bottom symmetry! Circles vs. Functions
    4. Owlcation: Vertical Line Test Examples
  4. Parabolas - These U-shaped graphs (like y=x²) always pass the Vertical Line Test since each x shoots to exactly one y. Whether you flip them, shift them, or stretch them, they stay function-friendly. Parabolas rock the single-output rule every time! Parabolas as Functions
    5. GeeksforGeeks: Vertical Line Test
  5. Ellipses & Hyperbolas - Ellipses and hyperbolas can be tricksters: vertical lines often hit them twice, so they're not functions by default. Watch out for both halves of the shape popping up around the axis. You might split them into pieces to make function-friendly sections! Conic Sections Quick Tip
    6. Owlcation: Vertical Line Test Examples
  6. Piecewise Functions - When you glue different rules together by x-intervals, you get piecewise functions. They pass the Vertical Line Test so long as each slice behaves itself with one y per x. Perfect for modeling real-world situations with changing rules! Piecewise Functions Breakdown
    7. Owlcation: Vertical Line Test Examples
  7. Vertical Lines - Ironically, vertical lines like x=3 fail their own test - they hit themselves infinite times! That means they aren't functions of x. Always avoid verticals when defining y as a function of x. About Vertical Lines
    8. Socratic: Vertical Line Test
  8. Horizontal Lines - Horizontal lines (y=5, y=-2, etc.) breeze through the test: each x maps to the same single y. They make for constant functions - super simple and dependable. Spot them by zero slope! Horizontal Lines Explained
    9. GeeksforGeeks: Vertical Line Test
  9. Absolute Value Functions - The classic V-shape of y=|x| always passes the Vertical Line Test. Each x fires off to precisely one y, even if two mirror-image arms appear. Absolute value functions are function-fans for sure! Absolute Value Insights
    10. GeeksforGeeks: Vertical Line Test
  10. Step Functions - Think of a staircase: step functions jump from one y-value to the next without breaking the one-output rule. Each x lands on one flat segment, so they're functions. Great for discrete modeling tasks like rounding or bucket rates! Step Functions Guide
    11. Owlcation: Vertical Line Test Examples
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