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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Which Graph is y = 1/2x²

Sharpen your skills with guided graph analysis

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Pick the Parabola trivia quiz for high school algebra students.

Which of the following equations represents a parabola opening upwards?
y = x^2
y = -x^2
x = y^2
y = 2
The equation y = x^2 is in the standard form of a parabola that opens upwards because its quadratic term has a positive coefficient. The other options either open in a different direction or are not parabolic equations.
What is the vertex of the parabola given by y = 1/2 x^2?
(0, 1/2)
(1/2, 0)
(-1/2, 0)
(0, 0)
For any quadratic function in the form y = ax^2 with no horizontal or vertical shifts, the vertex is at (0, 0). This is because there is no h or k translation in the equation.
In the equation y = 1/2 x^2, what effect does the coefficient 1/2 have on the parabola?
It makes the parabola narrower
It reflects the parabola
It makes the parabola wider
It shifts the vertex upward
A coefficient between 0 and 1 causes a vertical stretch that is less intense than the standard parabola, making it appear wider. No reflections or translations occur with this change in coefficient.
Which graph feature is common to all parabolas of the form y = ax^2 + bx + c?
They are symmetric about a vertical axis
They always have the y-intercept at the origin
They have a horizontal axis of symmetry
They always open upwards
Every quadratic function of the given form has a vertical line of symmetry that passes through its vertex. This inherent symmetry is one of the defining properties of parabolic graphs.
If a parabola opens upward, what does this indicate about the value of 'a' in y = ax^2?
a is non-real
a is positive
a is negative
a is zero
A positive leading coefficient ensures that the parabola opens upwards. This is a direct result of the quadratic term contributing positively to y as x moves away from the vertex.
How can the vertex of a quadratic equation in standard form y = ax^2 + bx + c be found?
By using the formula (-b/(2a), f(-b/(2a)))
By factoring the constant term
By setting x equal to c
By differentiating twice
The vertex of a quadratic function in standard form can be determined by using the formula x = -b/(2a) and then calculating y by substituting this value back into the equation. This formula provides a quick and reliable method for finding the vertex.
What is the axis of symmetry for the parabola y = 2x^2 - 4x + 1?
x = 1
x = -1
y = 1
y = -1
Using the formula x = -b/(2a), we find that the axis of symmetry is x = 4/(2*2) = 1. This vertical line splits the parabola into two mirror-image halves.
When a quadratic function is written as y = a(x - h)^2 + k, what do h and k represent?
h is the focus and k is the directrix
h and k are the intercepts
h and k are the coefficients of x and the constant term
h and k are the x and y coordinates of the vertex, respectively
In vertex form, the constants h and k shift the graph horizontally and vertically, respectively, positioning the vertex at the point (h, k). This form makes it very clear where the vertex is located.
Which transformation does the equation y = (x - 3)^2 + 2 represent compared to y = x^2?
A horizontal stretch
A shift left by 3 and down by 2
A reflection across the x-axis
A shift right by 3 and up by 2
The term (x - 3) indicates a horizontal shift 3 units to the right, and the +2 indicates a vertical shift 2 units upward. These transformations are straightforward when the equation is in vertex form.
For the quadratic equation y = -3x^2 + 6x - 2, what does the negative coefficient imply?
The graph has no axis of symmetry
The parabola opens downward
The parabola opens upward
The vertex is a minimum
A negative coefficient for the x^2 term means the parabola opens downward. This also indicates that the vertex represents the maximum point of the parabola.
What is the effect of increasing the absolute value of 'a' in y = ax^2 on the graph of the parabola?
The parabola becomes wider
The parabola becomes narrower
The parabola shifts horizontally
The vertex moves upward
A larger absolute value of 'a' makes the parabola steeper and narrower because the rate at which y increases (or decreases) with x becomes more rapid. This change affects the overall 'width' of the graph.
Which of the following represents the effect of reflecting a parabola across the x-axis?
y = -ax^2
y = a(x - h)^2 + k
y = ax^2 + c
x = -ay^2
Multiplying the quadratic term by -1, as shown in y = -ax^2, results in the graph being reflected over the x-axis. The reflection changes the opening direction of the parabola.
How many x-intercepts can a parabola have at most?
3
1
0
2
Since a quadratic is a second-degree equation, it can have at most two real roots, which correspond to the x-intercepts. Depending on the discriminant, it may have one or two intercepts.
Which quadratic form is most useful for identifying the vertex of the parabola?
Vertex form (y = a(x - h)^2 + k)
Standard form (y = ax^2 + bx + c)
Intercept form
Factored form (y = a(x - r1)(x - r2))
The vertex form clearly shows the vertex of the parabola as (h, k), making it the easiest form for identifying the vertex. Other forms require additional steps to extract vertex information.
In the quadratic function y = x^2 - 4x + 7, what is the location of the vertex?
(-2, -3)
(-2, 3)
(2, -3)
(2, 3)
The vertex can be found using x = -b/(2a). Here, x = 4/2 = 2, and substituting back yields y = (2)^2 - 4(2) + 7 = 3. Thus, the vertex is located at (2, 3).
Given the quadratic function y = 1/2 x^2 + 4x + 4, how do you determine its axis of symmetry?
Use the formula x = -b/(2a) to get x = -4
Assume the axis of symmetry is x = 0
Factor the quadratic to find the roots
Use the symmetry of the y-intercept
By applying the formula x = -b/(2a) with a = 1/2 and b = 4, we get x = -4/(2*(1/2)) = -4. This is the line about which the parabola is symmetric.
How do you complete the square to write y = x^2 + 6x + 5 in vertex form?
Express it as y = (x + 3)^2 + 5
Express it as y = (x - 3)^2 + 5
Express it as y = (x - 3)^2 - 4
Express it as y = (x + 3)^2 - 4
To complete the square, take half of 6 to get 3, then square it (9), and rewrite the quadratic as (x + 3)^2 - 9 + 5, which simplifies to (x + 3)^2 - 4. This method directly produces the vertex form.
For the quadratic equation y = -2x^2 + 8x - 3, what does the vertex represent?
It indicates the parabola has no maximum or minimum
It is the point of symmetry only
It represents the maximum point of the parabola
It represents the minimum point of the parabola
Since the leading coefficient is negative, the parabola opens downward, making the vertex its highest point, i.e. the maximum. This property is essential in optimization problems involving quadratics.
Which of the following methods can be used to determine the number of real x-intercepts of a quadratic function?
Converting to vertex form
Analyzing the discriminant (b^2 - 4ac)
Checking the coefficient of x^2
Graphing the y-intercept
The discriminant of a quadratic equation determines the number of real roots. A positive discriminant indicates two x-intercepts, zero signifies a single intercept, and a negative value means no real intercepts.
Consider the quadratic function y = ax^2. If the graph of this function is stretched vertically and then shifted upwards, which of the following represents this transformation?
y = a(x - k)^2, where k > 0
y = k + ax^2, where k > 0
y = a(x^2 + k)
y = a(x + k)^2 - k
A vertical stretch is applied by multiplying x^2 by a constant a, and an upward shift is applied by adding a positive constant k, resulting in the transformation y = ax^2 + k. This order of operations directly illustrates how the graph is modified.
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Study Outcomes

  1. Identify key features of quadratic functions, including vertex, axis of symmetry, and intercepts.
  2. Analyze the shapes and orientations of parabolic graphs to differentiate between various quadratic equations.
  3. Apply transformation rules to determine the effect of coefficient changes on the graph's appearance.
  4. Match given quadratic equations with their corresponding parabolic graphs efficiently.
  5. Evaluate and verify graph selections to build confidence in test and exam preparation.

Quiz: Which Graph Represents y=1/2x^2? Cheat Sheet

  1. Understand the Standard Form - Every quadratic loves to show off its outfit f(x) = ax² + bx + c, where a, b, and c are constants that shape your parabola's height and direction. Study Standard Form
  2. Identify Parabola Direction - A positive "a" makes your curve smile upward, while a negative "a" gives it a frown downward. Recognizing this helps you predict how your graph behaves at the edges. Learn Parabola Direction
  3. Locate the Vertex - The special point x = -b/(2a) is your parabola's peak or valley. Plug that back into f(x) to get the full VIP coordinates of your quadratic's turning point. Find the Vertex
  4. Draw the Axis of Symmetry - This vertical line x = -b/(2a) slices your parabola into two perfect mirror halves. Use it to plot matching points and make your sketch flawless. Axis of Symmetry Tips
  5. Find the Y‑Intercept - Simply evaluate f(0) to discover where your parabola crosses the y‑axis. It's the quickest free point on your graph to kickstart your plotting. Y‑Intercept Basics
  6. Solve for X‑Intercepts - Set f(x)=0 and factor, complete the square, or unleash the quadratic formula x = (-b ± √(b²-4ac))/(2a) to nail down your roots. Solve for Roots
  7. Master the Discriminant - The secret Δ = b²-4ac tells you if you've got two, one, or zero real roots. Let it guide you before you even crunch numbers. Discriminant Magic
  8. Explore Transformations - Shift up/down by adding constants, slide left/right inside the square, or flip your curve by swapping the sign of a. Play stylist with your parabola's look. Parabola Transformations
  9. Practice Graphing - Plot the vertex, axis, intercepts, and then connect the dots with a smooth U-shape. This hands‑on approach cements your grasp of quadratic behavior. Graphing Parabolas Guide
  10. Apply to Real‑Life Problems - From launching projectiles to maximizing areas, quadratics pop up everywhere. Tackling real scenarios brings your math skills into the real world. Quadratics in Real Life
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