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Quizzes > High School Quizzes > Mathematics

Quadratic Functions & Transformations Practice Quiz

Engage with Quadratic Function Worksheet Exercises

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting a Transformative Quadratics trivia quiz for high school students.

What is the vertex form of a quadratic function?
y = ax^2 + bx + c
y = a(x - h)^2 + k
y = (x - h)^2
y = a(x + h)^2 + k
The vertex form y = a(x - h)^2 + k directly shows the vertex (h, k) of the parabola. It makes it easier to identify transformations of the quadratic function.
Which of the following represents a vertical translation of a quadratic function?
y = f(x) + k
y = f(x)/k
y = f(x - k)
y = k*f(x)
Adding a constant outside the function, as in y = f(x) + k, shifts the graph vertically. The other options affect the graph in different ways.
What is the axis of symmetry for the quadratic function y = ax^2 + bx + c?
x = -b/(2a)
x = b/(2a)
x = -c/(2a)
x = a/(2b)
For any quadratic function in standard form, the axis of symmetry is given by x = -b/(2a). This line divides the parabola into two mirror images.
If a quadratic function has a > 0, how does its graph open?
Leftward
Upward
Downward
Rightward
When a > 0, the parabola opens upward, making the vertex a minimum point. This is a fundamental property of quadratic functions.
Which form of a quadratic function directly displays the y-intercept when x = 0?
Vertex form, y = a(x - h)^2 + k
Derivative form
Standard form, y = ax^2 + bx + c
Factored form, y = a(x - r)(x - s)
In the standard form, the constant c is the y-intercept because when x = 0, y equals c. Vertex and factored forms do not reveal the y-intercept directly.
How do you find the vertex of a quadratic function given in standard form y = ax^2 + bx + c?
Factor the quadratic
Use the discriminant formula
Set y = 0 and solve for x
Calculate x = -b/(2a) and substitute back to find y
The vertex's x-coordinate is found using -b/(2a), and substituting that value back gives the y-coordinate. This method directly derives the vertex from the standard form.
Which change in the equation y = a(x - h)^2 + k represents a horizontal shift to the right by 3 units?
Setting k = -3
Setting h = 3
Setting k = 3
Setting h = -3
In vertex form, the term (x - h) indicates a horizontal shift. When h is set to 3, the graph shifts 3 units to the right.
For the quadratic function f(x) = 2(x + 4)^2 - 5, what is the vertex?
(4, 5)
(-4, -5)
(4, -5)
(-4, 5)
The vertex form f(x) = 2(x + 4)^2 - 5 indicates a vertex at (-4, -5), since the expression x + 4 implies a horizontal shift by -4 and the constant -5 gives the vertical translation.
What is the effect of multiplying a quadratic function by a negative number?
It reflects the graph over the x-axis
It shifts the graph horizontally
It only stretches the graph vertically
It has no effect on the graph
Multiplying the function by a negative number reflects the graph across the x-axis, changing the direction in which the parabola opens.
How does the graph of y = (x - 2)^2 + 3 change when the equation is modified to y = 3(x - 2)^2 + 3?
It is horizontally stretched by a factor of 3
It is reflected across the x-axis
It shifts to the right by 3 units
It is vertically stretched by a factor of 3
Multiplying the squared term by 3 stretches the parabola vertically, making it narrower. The vertex remains unchanged because the horizontal and vertical shifts are not affected.
Given a quadratic function in standard form y = ax^2 + bx + c, what role does the coefficient a play?
It determines the direction and width of the parabola
It provides the line of symmetry
It determines the x-coordinate of the vertex
It only affects the vertical translation
The coefficient a dictates whether the parabola opens upward (a > 0) or downward (a < 0) and affects its width or steepness. The other coefficients influence different aspects of the graph.
How can you determine the number of real roots of a quadratic equation?
By averaging the coefficients
By calculating the discriminant (b^2 - 4ac)
By rewriting it in vertex form
By completing the square
The discriminant b^2 - 4ac reveals the nature and number of roots: a positive value indicates two real roots, zero indicates one, and a negative value indicates complex roots. This is a standard method for assessing quadratic solutions.
If a quadratic function is written in factored form as y = a(x - r)(x - s), what do r and s represent?
They are the coordinates of the vertex
They are the x-intercepts of the function
They indicate the maximum and minimum values
They are the y-intercepts
In factored form, setting (x - r) or (x - s) equal to zero gives the x-intercepts where the graph crosses the x-axis. These values are key to understanding the function's behavior.
When graphing quadratic functions, what effect does adding a constant to the equation have?
It reflects the graph over the y-axis
It stretches the graph horizontally
It translates the graph vertically
It translates the graph horizontally
Adding a constant outside the quadratic function causes a vertical translation, shifting the graph up or down without altering its shape.
Which method is used to convert a quadratic function from standard form to vertex form?
Factoring
Graphing
Differentiation
Completing the square
Completing the square is a systematic method to rewrite a quadratic equation in vertex form. This conversion makes it easier to identify the vertex and analyze transformations.
Consider the quadratic function y = -3(x - 1)^2 + 4. What is the equation of its axis of symmetry and its vertex?
Axis of symmetry: x = 3, Vertex: (3, 4)
Axis of symmetry: x = -1, Vertex: (-1, 4)
Axis of symmetry: x = 1, Vertex: (1, 4)
Axis of symmetry: y = 4, Vertex: (1, -3)
The vertex form immediately shows the vertex (1, 4) and indicates that the axis of symmetry is the line x = 1. The negative coefficient affects the opening direction but not the location of the vertex or axis.
A quadratic function undergoes the transformation y = 0.5f(2(x + 1)) - 3 from the base function y = f(x). Which of the following describes the sequence of transformations correctly?
Horizontal compression by a factor of 2, shift left by 1, vertical shrink by a factor of 0.5, and shift down by 3
Horizontal stretch by a factor of 2, shift right by 1, vertical shrink by a factor of 0.5, and shift up by 3
Horizontal compression by a factor of 1/2, shift left by 1, vertical shrink by a factor of 0.5, and shift down by 3
Horizontal compression by a factor of 1/2, shift left by 1, vertical shrink by a factor of 2, and shift down by 3
The multiplier 2 inside the function causes a horizontal compression by a factor of 1/2, and adding 1 inside the parentheses results in a left shift by 1. The external multiplier of 0.5 shrinks the graph vertically, and subtracting 3 shifts it downward.
When converting the quadratic equation 2x^2 + 8x + 5 to vertex form by completing the square, what is the correct vertex form?
2(x - 2)^2 + 3
2(x - 2)^2 - 3
2(x + 2)^2 + 3
2(x + 2)^2 - 3
Factoring out 2 and completing the square for 2x^2 + 8x + 5 results in 2[(x + 2)^2 - 4] + 5, which simplifies to 2(x + 2)^2 - 3. This vertex form shows the vertex clearly.
Determine the values of a, h, and k in the vertex form y = a(x - h)^2 + k for the quadratic function given by y = -4x^2 + 16x - 15.
a = -4, h = 2, k = -1
a = 4, h = 2, k = -1
a = -4, h = 2, k = 1
a = -4, h = -2, k = 1
By completing the square on y = -4x^2 + 16x - 15, you can rewrite it as -4(x - 2)^2 + 1. This gives a = -4, h = 2, and k = 1, which are derived directly from the vertex form.
If the graph of a quadratic function y = ax^2 + bx + c is reflected over its axis of symmetry, what is the resulting graph?
It is reflected over the y-axis
It becomes a linear function
It remains unchanged
It is reflected over the x-axis
Because a quadratic function is symmetric about its axis, reflecting it over that axis produces an identical graph. This is a unique property of symmetric functions.
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Study Outcomes

  1. Identify the standard form and vertex form of quadratic functions.
  2. Analyze the effects of translations, reflections, and dilations on quadratic graphs.
  3. Apply transformation rules to convert between different representations of quadratic functions.
  4. Evaluate the impact of parameter changes on the graph's shape and position.
  5. Solve problems involving quadratic transformations to predict and sketch function behavior.

Quadratic Functions & Transformations Worksheet Cheat Sheet

  1. Understand the vertex form - Vertex form of a quadratic is f(x)=a(x−h)2+k, where (h,k) marks the parabola's tip. It instantly tells you shifts and vertical stretches or shrinks, so you can graph faster than ever. Mastering this makes transforming quadratics feel like unlocking a secret code! OpenStax: Graph Quadratic Functions
  2. Learn how the coefficient a affects the graph - The value of a controls the parabola's "skinny or chubby" factor: if |a|>1 it squeezes inward, and if 0<|a|<1 it balloons outward. Flip the sign to watch the curve dive below the x-axis instead of soaring above. Think of it as tuning a slinky - change the tension and the shape springs to life! OpenStax: Graph Quadratic Functions
  3. Practice vertical shifts - Adding k in f(x)=x2+k moves the entire graph up by k units; subtracting k drags it down. It's like playing a video game character on an elevator - dial the number and you're slider-floating to a new height. Try different values to build intuition for how every vertical tweak matters! OpenStax: Graph Quadratic Functions
  4. Master horizontal shifts - Replacing x with (x−h) shifts the parabola right by h units; using (x+h) slides it left. Picture dragging a lamp along a table - every h-unit click moves your spotlight. Getting comfortable here means no more guessing where your curve will land on the x-axis! OpenStax: Graph Quadratic Functions
  5. Combine transformations - In f(x)=a(x−h)2+k you stretch or compress by a, slide horizontally by h, and lift vertically by k all at once. It's like mixing colors on a palette - each parameter blends to give a unique curve. Practice layering these changes to become a true transformation maestro! OpenStax: Graph Quadratic Functions
  6. Identify the axis of symmetry - The vertical line x=h cuts the parabola into mirror‑image halves. This axis helps you reflect any plotted point to find its twin, making sketches faster and more accurate. Once you spot it, the rest of the graph practically draws itself! OpenStax: Graph Quadratic Functions
  7. Recognize the vertex - The vertex is the peak or valley of your parabola, sitting at (h,k). It's the absolute max or min, so it tells you exactly where your function hits its highest or lowest score. Finding this superstar point first makes the whole graphing mission a breeze! OpenStax: Graph Quadratic Functions
  8. Understand the direction of opening - If a>0 the parabola opens upward like a smiling U; if a<0 it flips into a frowning ∩. That little positive or negative sign tells you whether you're cheering for a win or bracing for a drop. Always check this first so you know your curve's mood! OpenStax: Graph Quadratic Functions
  9. Practice completing the square - Convert ax2+bx+c into vertex form by adding and subtracting (b/2a)2. This algebraic maneuver reveals h and k, unlocking all your transformation powers. It's a must-have skill for conquering quadratics and boosting your graphing confidence! OpenStax: Graph Quadratic Functions
  10. Use graphing tools - Digital grapher apps or online calculators let you slide parameters a, h, and k in real time. Watching your parabola dance as you tweak values cements the connection between equations and visuals. Mix practice on paper with tech tools to become unstoppable in graphing quadratics! OpenStax: Graph Quadratic Functions
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