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Quizzes > High School Quizzes > Mathematics

Graphing Radical Functions Practice Quiz

Sharpen skills using step-by-step radical graphing problems

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Radical Graphing Challenge quiz art for high school math students to practice graphing functions.

What is the domain of the function y = √x?
x ≥ 0
x ≤ 0
All real numbers
x > 0
The square root function is only defined when the radicand is nonnegative. Therefore, x must be greater than or equal to 0 for y = √x to be valid.
What is the range of the function y = √x?
y ≥ 0
y ≤ 0
All real numbers
y > 0
Since the square root of any nonnegative number is nonnegative, the function y = √x produces outputs that are greater than or equal to 0. Thus, the range is y ≥ 0.
How does the graph of y = √(x - 3) differ from the graph of y = √x?
It shifts 3 units to the right
It shifts 3 units to the left
It shifts 3 units upward
It stretches horizontally by a factor of 3
Replacing x with (x - 3) inside the square root causes a horizontal translation of the graph 3 units to the right. The overall shape remains the same, only its position changes.
What is the effect of adding 2 to the function y = √x, as in y = √x + 2?
The graph shifts upward by 2 units
The graph shifts downward by 2 units
The graph shifts to the right by 2 units
There is no shift; only the output values increase
Adding 2 to the function moves every output value up by 2. This results in a vertical translation of the graph upward by 2 units.
How does the graph of y = -√x differ from y = √x?
It reflects the graph over the x-axis
It reflects the graph over the y-axis
It shifts the graph upward
It shifts the graph downward
The negative sign in front of the square root function multiplies every output by -1, reflecting the graph across the x-axis. The domain remains the same but all y-values become non-positive.
What is the domain of the function y = √(2x + 6)?
x ≥ -3
x ≤ -3
x > 3
All real numbers
The radicand 2x + 6 must be greater than or equal to 0. Solving 2x + 6 ≥ 0 yields x ≥ -3, which is the domain of the function.
Which transformation describes the function y = -√(x + 2) - 3?
Left 2 units, reflected over the x-axis, and shifted down 3 units
Right 2 units, reflected over the x-axis, and shifted up 3 units
Left 2 units and shifted up 3 units
Reflected over the y-axis and shifted down 3 units
The term (x + 2) inside the square root shifts the graph 2 units to the left. The negative sign reflects the graph over the x-axis and subtracting 3 shifts it downward by 3 units.
What is the range of the function y = -√(x - 1) + 4?
y ≤ 4
y ≥ 4
y ≥ 0
y ≤ 0
The basic function y = √(x - 1) has a minimum output of 0; reflecting it makes the outputs non-positive. Adding 4 shifts the maximum to 4, so the range is all y-values less than or equal to 4.
Which function represents a horizontal stretch of y = √x by a factor of 4?
y = √(x/4)
y = 4√x
y = √(4x)
y = (√x)/4
A horizontal stretch by a factor of 4 is achieved by replacing x with x/4 inside the square root. This transformation spreads the graph out horizontally without altering its vertical scale.
What is the inverse of y = √x + 2?
f❻¹(x) = (x - 2)² for x ≥ 2
f❻¹(x) = √(x - 2) for x ≥ 2
f❻¹(x) = (x + 2)² for x ≥ -2
f❻¹(x) = √(x + 2)
To find the inverse, swap x and y to form x = √y + 2. By isolating the square root and then squaring both sides, the inverse function is determined to be f❻¹(x) = (x - 2)² with the restriction that x ≥ 2.
Find the x-intercept of the function y = 2√(x - 5) - 3.
7.25
5
3
2.25
Setting y equal to 0 gives 2√(x - 5) - 3 = 0, which simplifies to √(x - 5) = 1.5. Squaring both sides yields x - 5 = 2.25, so the x-intercept is x = 7.25.
What transformation does the graph of y = 3√x undergo compared to y = √x?
Vertical stretch by a factor of 3
Horizontal stretch by a factor of 3
Vertical shift upward by 3 units
Horizontal shift right by 3 units
Multiplying the square root function by 3 scales all output values by 3, resulting in a vertical stretch. The shape of the graph remains similar, but it becomes taller.
Determine the effect of the transformation in y = -2√(x + 4) - 1 compared to y = √x.
It shifts left 4 units, reflects over the x-axis, vertically stretches by a factor of 2, and shifts down 1 unit
It shifts right 4 units, reflects over the y-axis, vertically stretches by a factor of 2, and shifts up 1 unit
It shifts left 4 units, vertically compresses by a factor of 2, and shifts up 1 unit
It reflects over the x-axis, vertically stretches by a factor of 2, and shifts up 4 units
The expression (x + 4) shifts the graph 4 units to the left, the coefficient -2 reflects the graph over the x-axis and stretches it vertically by a factor of 2, and subtracting 1 shifts it downward by 1 unit.
If y = √x is translated so that its vertex is at (2, -3), what is the equation of the function?
y = √(x - 2) - 3
y = √(x + 2) - 3
y = √(x - 2) + 3
y = √(x + 2) + 3
To shift the vertex to (2, -3), the function must be horizontally translated by replacing x with (x - 2) and then shifted downward by subtracting 3. This gives the equation y = √(x - 2) - 3.
Determine the range of the function y = 4 - √(5x + 10).
y ≤ 4
y ≥ 4
y ≥ 0
y ≤ 0
The maximum value of y occurs when the square root term is zero, which happens when 5x + 10 = 0. At that point, y = 4. For all other values of x, the square root is positive, making y less than 4.
Solve the radical equation √(2x - 1) + √(x - 1) = 3.
x = 27 - 2√153
x = 27 + 2√153
x = 2
No solution
By isolating one of the square root terms and squaring both sides, you eventually obtain a quadratic equation. After solving and checking for extraneous solutions, the only valid solution is x = 27 - 2√153, which is approximately 2.26.
Which of the following equations represents a horizontally compressed and vertically shifted radical function, starting from y = √x, such that the vertex is at (0, -2) and the graph is compressed horizontally by a factor of 1/2?
y = √(2x) - 2
y = √(x/2) - 2
y = √(2x - 2)
y = 2√x - 2
A horizontal compression by a factor of 1/2 is achieved by replacing x with 2x inside the square root, resulting in y = √(2x). Subtracting 2 then shifts the graph downward so that the vertex is at (0, -2).
For the function y = -√(x - 4) + 6, determine the coordinates of its vertex.
(4, 6)
(-4, 6)
(4, -6)
(-4, -6)
The vertex of a radical function in the form y = -√(x - h) + k is found where the expression inside the square root is zero, which is at x = h. Therefore, the vertex is at (4, 6), regardless of the reflection.
If a radical function is defined as y = a√(b(x - c)) + d and passes through the point (c, d), which of the following must be true?
The vertex of the function is (c, d)
The function has a y-intercept at (c, d)
The function has a horizontal asymptote at y = d
The function is symmetric about the line x = c
When x = c, the expression under the square root becomes zero, yielding y = d. This point (c, d) is the vertex of the function, which is a characteristic feature of this form.
Consider the function y = √((x+3)²) - 5. Which of the following is equivalent to this function for all x, and what does its graph describe?
y = |x+3| - 5, which is a V-shaped graph
y = x+3 - 5, a straight line
y = |x| - 8, a V-shaped graph
y = √x - 2, a half-parabolic graph
The expression √((x+3)²) simplifies to the absolute value |x+3|. Therefore, the function can be rewritten as y = |x+3| - 5, which describes a V-shaped graph with its vertex at (-3, -5).
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Study Outcomes

  1. Understand the properties and characteristics of radical functions.
  2. Graph radical functions accurately based on their transformations.
  3. Analyze the effects of translations, reflections, and dilations on radical graphs.
  4. Interpret domain and range restrictions of radical functions.
  5. Apply instantaneous feedback to improve graphing techniques and problem-solving strategies.

Graphing Radical Functions Cheat Sheet

  1. Master the Parent Function - Every radical explorer starts with the parent function f(x) = √x, which lays the foundation for all the cool transformations you'll encounter. It maps out the basic curve and the point where your graph springs to life. Symbolab: Graph Radical Functions
  2. Find the Domain - To keep your graphs in the realm of real numbers, ensure the radicand (the expression under the root) is never negative. Think of it as setting boundaries so your graph knows exactly where it's allowed to roam. OnlineMathLearning: Domain of Radical Equations
  3. Explore Vertical Shifts - Adding or subtracting a constant outside the radical scoots your graph up or down, like giving it elevator rides. It's a quick way to adjust the starting height and see how the curve dances on the y‑axis. Symbolab: Radical Function Transformations
  4. Investigate Horizontal Shifts - Tucking a constant inside the radical, as in √(x - h), slides the entire graph left or right by h units. It's like watching your graph take steps along the x‑axis - super handy for lining up important features. Symbolab: Radical Function Transformations
  5. Understand Reflections - Multiply by - 1 in front of your radical to flip the graph over the x‑axis, giving it a cool mirror‑image effect. This reflection trick instantly changes the orientation and can spice up your curve's look. Symbolab: Radical Function Reflections
  6. Practice with Value Tables - Craft a table of values - pick x's that make perfect squares under the root so you can calculate points in a snap. This hands‑on plotting drill turns your abstract function into a clear, rock‑solid picture. Symbolab: Graph Radical Functions
  7. Spot the Range - The range springs from the lowest y‑value your transformations allow, and it tells you how high or low your graph can go. Tracking vertical moves, stretches, and flips will give you the full picture of possible outputs. OnlineMathLearning: Range of Radical Equations
  8. Try Vertical Stretch & Shrink - Multiply by a constant >1 to stretch your graph taller or by a number between 0 and 1 to shrink it shorter. It's the secret sauce for tweaking the curve's steepness and making it pop. OnlineMathLearning: Stretching Radical Graphs
  9. Use Graphing Tech - Fire up GeoGebra or your favorite graphing software to play with transformations in real time. Interactive tools help you visualize shifts, stretches, and flips faster than you can say "radical!" GeoGebra: Radical Function Simulator
  10. Engage with Practice Problems - Reinforce your new skills by tackling quizzes and worksheets that mix domain, range, and all transformations. The more you practice, the more confident you'll become at sketching radical functions on demand. MathBitsNotebook: Radical Practice
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