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Quizzes > High School Quizzes > Mathematics

Practice Quiz: Graphing Quadratic Functions

Test your graphing skills through guided practice

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Graphing Quadratic Challenge quiz for high school algebra students.

What is the standard form of a quadratic function?
y = ax^2 + bx + c
y = a(x - h)^2 + k
y = (x - h)^2 + c
y = a(x + h)^2 + k
The standard form of a quadratic function is written as y = ax^2 + bx + c, where a, b, and c are constants. This form is useful for identifying the y-intercept and performing further calculations.
In the vertex form of a quadratic function, y = a(x − h)^2 + k, what does the point (h, k) represent?
The vertex of the parabola
The x-intercept
The axis of symmetry
The focus
The vertex form clearly shows that the vertex of the quadratic function is at (h, k). This form makes it easy to identify the maximum or minimum point of the parabola.
What role does the coefficient 'a' play in the quadratic function y = ax^2 + bx + c?
It determines the direction and width of the parabola
It controls the vertical shift
It represents the x-coordinate of the vertex
It determines the y-intercept
The coefficient 'a' affects whether the parabola opens upwards or downwards and influences how narrow or wide the graph appears. A larger absolute value of 'a' results in a narrower parabola.
Where does the quadratic function y = ax^2 + bx + c intersect the y-axis?
(0, c)
(c, 0)
(a, c)
(b, c)
The y-intercept of any function is found by setting x = 0. For the quadratic function, substituting x = 0 yields y = c, so the intersection point is (0, c).
If the leading coefficient 'a' in the quadratic function is positive, which direction does the parabola open?
Upward
Downward
Sideways to the right
Sideways to the left
A positive leading coefficient means that the parabola opens upward, forming a U-shaped graph. This implies that the vertex is the minimum point on the graph.
How do you find the axis of symmetry for a quadratic function in standard form y = ax^2 + bx + c?
x = -b/(2a)
x = -c/(2a)
x = -b/a
x = c/(2a)
The axis of symmetry for a quadratic function in standard form is given by the formula x = -b/(2a). This line divides the parabola into two mirror-image halves.
What does the discriminant, found in the expression b^2 - 4ac, indicate about a quadratic equation?
It indicates the number and nature of the roots
It shows the vertex of the parabola
It determines the y-intercept
It provides the factorization of the quadratic
The discriminant helps determine whether the quadratic equation has two distinct real roots, one real repeated root, or two complex roots. A positive discriminant indicates two real roots, zero indicates one, and a negative discriminant indicates complex roots.
What is the vertex of the quadratic function y = 2x^2 - 8x + 5?
(2, -3)
(-2, 1)
(4, 5)
(-4, 5)
To find the vertex, use the formula h = -b/(2a) which gives h = 2, and then substitute back into the equation to find k = -3. Thus, the vertex is (2, -3).
Which method is most effective for rewriting y = x^2 + 6x + 8 in vertex form?
Completing the square
Factoring
Using the quadratic formula
Graphing directly
Completing the square is a reliable method to convert a quadratic in standard form to vertex form. It reorganizes the equation to clearly show the vertex and the shape of the parabola.
For the function y = -3x^2 + 12x - 7, what does the vertex represent on the graph?
The maximum point
The minimum point
The y-intercept
The point of inflection
Because the quadratic has a negative leading coefficient, the parabola opens downward. Therefore, the vertex represents the maximum point on the graph.
How does increasing the absolute value of 'a' in y = a(x - h)^2 + k affect the parabola's graph?
It makes the parabola narrower
It makes the parabola wider
It shifts the parabola horizontally
It does not affect the graph's shape
A larger absolute value of 'a' causes the parabola to be steeper and narrower. This affects the rate at which the function's values increase or decrease away from the vertex.
Which property of a parabola remains unchanged when the function undergoes a vertical shift?
The shape and the width of the parabola
The y-intercept
The vertex's position
The x-intercepts
A vertical shift moves the parabola up or down but does not alter its shape or width. The coefficient 'a' remains unchanged, preserving the graph's steepness and overall appearance.
What does the discriminant of a quadratic function reveal about its roots?
It reveals the number and type of the roots (real or complex)
It indicates the vertex's coordinates
It shows how the graph is shifted vertically
It determines the axis of symmetry
The discriminant, given by b^2 - 4ac, tells us whether the roots are real or complex and whether they are distinct or repeated. This is a key component in analyzing the quadratic function's solutions.
How is the graph of y = (x - 1)^2 + 4 described?
A parabola with vertex (1, 4) opening upward
A parabola with vertex (-1, 4) opening downward
A parabola with vertex (1, -4) opening upward
A parabola with vertex (-1, -4) opening upward
The given function is in vertex form, which directly shows that the vertex is at (1, 4). Since the coefficient a is positive (implicitly 1), the parabola opens upward.
What is the simplest method to find the x-intercepts of y = x^2 - 5x + 6?
Factoring the quadratic
Completing the square
Using the quadratic formula
Graphing the function
The quadratic y = x^2 - 5x + 6 factors easily into (x - 2)(x - 3) = 0, which quickly yields the x-intercepts. Factoring is typically the simplest method when the quadratic can be factored.
For the quadratic function y = 4x^2 - 16x + 15, what is the vertex?
(2, -1)
(4, 15)
(-2, -15)
(2, 1)
To find the vertex, compute h = -b/(2a) = 16/(8) = 2, then substitute back into the equation to find k = -1. Thus, the vertex is (2, -1).
What is the axis of symmetry for the quadratic function y = -2(x + 3)^2 + 7?
x = -3
x = 3
x = 7
x = -7
The vertex form y = a(x - h)^2 + k shows that the x-coordinate of the vertex is h. In the function y = -2(x + 3)^2 + 7, rewriting (x + 3) as (x - (-3)) reveals that h is -3, so the axis of symmetry is x = -3.
For the quadratic function y = x^2 - 4x + k, what value of k results in a single x-intercept?
k = 4
k = -4
k = 0
k = 8
A quadratic function has a single x-intercept when its discriminant is zero. For the quadratic y = x^2 - 4x + k, setting the discriminant (16 - 4k) equal to zero yields k = 4, resulting in one repeated root.
How does reflecting a quadratic function across the x-axis modify its equation and graph?
It changes the sign of 'a' so that the parabola opens in the opposite direction
It changes the vertex's position
It alters the axis of symmetry
It shifts the graph upward
Reflecting a quadratic over the x-axis multiplies the function by -1, which changes the sign of the leading coefficient 'a'. This reflection causes the parabola to open in the opposite direction while retaining its shape.
If a quadratic function has a vertex at (2, -3) and passes through the point (4, 5), what is its equation in vertex form?
y = 2(x - 2)^2 - 3
y = -2(x - 2)^2 - 3
y = 2(x + 2)^2 - 3
y = 2(x - 2)^2 + 3
With the vertex (2, -3), the vertex form is y = a(x - 2)^2 - 3. Substituting the point (4, 5) into the equation gives 5 = a(2)^2 - 3, leading to a = 2. Thus, the equation is y = 2(x - 2)^2 - 3.
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Study Outcomes

  1. Analyze the structure of quadratic functions and identify components such as the vertex, axis of symmetry, and intercepts.
  2. Apply transformations to graph quadratic functions, including shifts and reflections.
  3. Interpret the effects of coefficient changes on the shape and orientation of quadratic graphs.
  4. Synthesize information from graphs to solve problems involving quadratic equations.

Graphing Quadratic Functions Practice Cheat Sheet

  1. Standard Form - Every quadratic starts life as f(x) = ax² + bx + c, where a, b, and c are your magic ingredients. Knowing this form helps you instantly spot whether your parabola smiles (opens up) or frowns (opens down) and how "wide" or "skinny" it looks. OpenStax Algebra & Trig
    2. openstax.org
  2. Vertex Formula - The vertex is the "peak" or "valley" of your parabola, and you find it with x = - b/(2a). Plug that back into f(x) to get the y-coordinate, and voilà - you've got the turning point! It's like plotting the highest jump or the deepest dip. OpenStax Algebra & Trig
    3. openstax.org
  3. Axis of Symmetry - Draw the vertical line x = - b/(2a) through your vertex, and you've split the parabola into two perfect mirror images. Think of it as your graph's runway centerline - everything on one side matches the other. Perfect symmetry makes sketching a breeze! OpenStax Algebra & Trig
    4. openstax.org
  4. Opening Direction - If a > 0, your parabola opens upward (happy face!), and if a < 0, it opens downward (sad face!). That little coefficient a is your "mood ring" for quadratics - flip it, and the graph flips. It's a quick mood check before you start plotting. OpenStax Algebra & Trig
    5. openstax.org
  5. Y‑Intercept - Set x = 0 in f(x) = ax² + bx + c, and you get f(0) = c, which is where the graph hits the y-axis. It's like plotting your starting point - ground zero for all those lovely curves. Pin it down first to anchor your sketch! OpenStax Algebra & Trig
    6. openstax.org
  6. X‑Intercepts (Roots) - Solve ax² + bx + c = 0 by factoring, completing the square, or using x = [ - b ± √(b² - 4ac)] / (2a). Those solutions are exactly where your parabola crosses the x-axis - perfect touchdown points! Roots reveal all the real-world intersection action. OpenStax Algebra & Trig
    7. openstax.org
  7. Vertex Form - Rewrite f(x) as a(x - h)² + k, where (h, k) is your vertex. This form is like tuning into the graph's broadcast - shifts and stretches jump off the page. Perfect for quick sketching or translating parabolas around the plane! OpenStax Algebra & Trig
    8. openstax.org
  8. Domain & Range - Quadratics live on all real x-values (domain = ( - ∞, ∞)). Their range depends on whether they open up [k, ∞) or down ( - ∞, k], with k from your vertex. It's like setting the stage: unlimited run for x, and y has a starting block at the vertex! OpenStax Algebra & Trig
    9. openstax.org
  9. Graphical Symmetry - Every point on one side of x = - b/(2a) has a twin on the other side with the same y-value. Imagine folding the paper in half along the axis of symmetry and watching the two halves line up perfectly. It's graph origami in action! OpenStax Algebra & Trig
    10. openstax.org
  10. Graphing Steps - Plot your vertex, draw the axis of symmetry, mark intercepts, then connect the dots with a smooth curve. Think of it as a 4‑step choreography for your parabola: locate, mirror, cross, and curve! Soon you'll be sketching like a graph wizard. OpenStax Algebra & Trig
    11. openstax.org
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