Graphing Quadratics — Drills
Graphing Quadratics has 10 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Factor — A number that divides evenly into another.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Parabola — The U-shaped graph of a quadratic function.
- Vertex — The turning point — the minimum if it opens up, the maximum if it opens down.
- Axis of symmetry — The vertical line through the vertex, x = -b2a.
- Roots — The x values where y = 0.
- Vertex form — y = a(x - h)2 + k, with vertex at (h, k).
- Height — The perpendicular distance from the base to the opposite corner.
Opens
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Does y = 1x2 + 3x + 8 open up or down?
Worked answer
up
a = 1
1 > 0, so it opens upward.
Problem 2
Does y = 5x2 + 0x - 1 open up or down?
Worked answer
up
a = 5
5 > 0, so it opens upward.
Axis
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the axis of symmetry of y = 5x2 + 50x + 2.
Worked answer
x = -5
-502(5) = -5
x = -5
Problem 2
Find the axis of symmetry of y = 1x2 - 8x + -2.
Worked answer
x = 4
--82(1) = 4
x = 4
Yint
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the y-intercept of y = 6x2 + 4x - 8.
Worked answer
-8
Every term with x vanishes.
y = -8
Problem 2
Find the y-intercept of y = 1x2 + 1x + 7.
Worked answer
7
Every term with x vanishes.
y = 7
Vertexform
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Give the vertex of y = 4(x - 0)2 - 5.
Worked answer
(0, -5)
x - 0 is zero at x = 0.
k = -5
Vertex (0, -5)
Problem 2
Give the vertex of y = (x + 3)2 + 8.
Worked answer
(-3, 8)
x + 3 is zero at x = -3.
k = 8
Vertex (-3, 8)
Vertex
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the vertex of y = x2 - 2x + 2.
Worked answer
(1, 1)
x = --22(1) = 1
y = 1(1)2 - 2(1) + 2 = 1
Vertex (1, 1), a minimum.
Problem 2
Find the vertex of y = 2x2 + 20x + 44.
Worked answer
(-5, -6)
x = -202(2) = -5
y = 2(-5)2 + 20(-5) + 44 = -6
Vertex (-5, -6), a minimum.
Roots
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the roots of y = x2 - 4x + 3.
Worked answer
1 and 3
(x - 1)(x - 3) = 0
x = 1 or x = 3
Problem 2
Find the roots of y = x2 + 11x + 18.
Worked answer
-9 and -2
(x + 9)(x + 2) = 0
x = -9 or x = -2
Width
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which parabola is narrowest?
- y = x2
- y = 6x2
- y = 0.1x2
- y = 0.5x2
Worked answer
y = 6x2
|6| is the largest coefficient.
So it rises fastest and is narrowest.
Minmax
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Does y = -5x2 + 2x + 1 have a maximum or a minimum?
Worked answer
maximum
a = -5, so it opens downward.
The vertex is the highest point, so a maximum.
Problem 2
Does y = -1x2 + 0x - 8 have a maximum or a minimum?
Worked answer
maximum
a = -1, so it opens downward.
The vertex is the highest point, so a maximum.
In context
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A ball's height is h(t) = -16t2 + 96t + 7 feet. Find the greatest height.
Worked answer
151
t = -962(-16) = 3 s
h(3) = -16(9) + 96(3) + 7
= 151 feet
Problem 2
A ball's height is h(t) = -16t2 + 32t + 5 feet. Find the greatest height.
Worked answer
21
t = -322(-16) = 1 s
h(1) = -16(1) + 32(1) + 5
= 21 feet
Symmetry
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A parabola has vertex (0, 1) and passes through (6, 1). Find another point on it.
Worked answer
(-6, 1)
The axis is x = 0.
(6, 1) is 6 to the right of it.
So (-6, 1) is 6 to the left.
Problem 2
A parabola has vertex (3, -2) and passes through (6, -4). Find another point on it.
Worked answer
(0, -4)
The axis is x = 3.
(6, -4) is 3 to the right of it.
So (0, -4) is 3 to the left.