Graphing Quadratics — Practice
Every parabola has a vertex, an axis of symmetry, and up to two roots. Find those and the graph draws itself.
What these questions 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.
Problem 1
Does y = 3x2 - 5x + 1 open up or down?
Worked answer
up
a = 3 > 0
So it opens upward
Problem 2
Does y = -2x2 + 7 open up or down?
Worked answer
down
a = -2 < 0
So it opens downward
Problem 3
Find the axis of symmetry of y = x2 - 6x + 2.
Worked answer
x = 3
--62(1) = 3
x = 3
Problem 4
Find the y-intercept of y = 2x2 + 3x - 8.
Worked answer
-8
Everything with x vanishes
y = -8
Problem 5
Give the vertex of y = (x - 4)2 + 3.
Worked answer
(4, 3)
h = 4, k = 3
(4, 3)
Problem 6
Give the vertex of y = 2(x + 5)2 - 7.
Worked answer
(-5, -7)
x + 5 means h = -5
(-5, -7)
Problem 7
Find the vertex of y = x2 - 8x + 11.
Worked answer
(4, -5)
x = 82 = 4
y = 16 - 32 + 11 = -5
Problem 8
Find the vertex of y = -x2 + 6x - 5.
Worked answer
(3, 4)
x = -62(-1) = 3
y = -9 + 18 - 5 = 4
Problem 9
Find the roots of y = x2 - 5x + 6.
Worked answer
2 and 3
(x-2)(x-3) = 0
x = 2 or x = 3
Problem 10
The roots of a parabola are -1 and 7. Find the axis of symmetry.
Worked answer
x = 3
-1 + 72 = 3
x = 3
Problem 11
Which parabola is narrowest?
- y = 5x2
- y = x2
- y = 0.5x2
- y = 0.1x2
Worked answer
y = 5x2
|5| is the largest
So y = 5x2 rises fastest and is narrowest
Problem 12
Does y = 4x2 - 8x + 1 have a maximum or a minimum?
Worked answer
minimum
The vertex is the lowest point
So a minimum
Problem 13
A ball's height is h(t) = -16t2 + 80t + 6 feet. Find the greatest height.
Worked answer
106
t = -802(-16) = 2.5 s
h(2.5) = -100 + 200 + 6 = 106 feet
Problem 14
For h(t) = -16t2 + 80t + 6, at what time is the height greatest?
Worked answer
2.5
t = 8032 = 2.5 seconds
Problem 15
A parabola has vertex (2, -3) and passes through (5, 6). Find another point on it.
Worked answer
(-1, 6)
(5, 6) is 3 right of the axis x = 2
So 3 left is x = -1
(-1, 6)
Problem 16
Find the roots of y = 2x2 - 8x.
Worked answer
0 and 4
2x(x - 4) = 0
x = 0 or x = 4
Never divide by x — it would lose the root 0