Graphing Quadratics — Print
Every parabola has a vertex, an axis of symmetry, and up to two roots. Find those and the graph draws itself.
What the practice 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.
Throw a ball and photograph it every half second.
The dots are not evenly spaced. Going up they bunch closer together; coming down they spread apart.
It slows as it rises, stops for an instant at the top, then speeds up on the way down.
Now fold the picture down the dashed line through the highest point. Every ball lands on its twin.
The path is perfectly symmetric — the way up is the way down, reversed.
That mirror line is the axis of symmetry, and the top of it is the vertex. Find those two and the rest of the parabola draws itself, because the left half is just the right half reflected.
Which way it opens
If a > 0 the parabola opens upward and the vertex is the lowest point — a minimum.
If a < 0 it opens downward and the vertex is a maximum.
A larger |a| makes the curve narrower; a smaller one makes it wider.
The sign of a decides
Positive a: happy face, minimum. Negative a: sad face, maximum. Check this before anything else — it tells you what kind of answer to expect.
Finding the vertex
From standard form, the axis of symmetry is x = -b2a. Substitute that x back in to get the y.
For y = x2 - 6x + 5: x = 62 = 3, and y = 9 - 18 + 5 = -4. Vertex (3, -4).
From vertex form y = a(x - h)2 + k, the vertex is simply (h, k) — read straight off.
Watch the sign in vertex form
y = 2(x - 5)2 + 3 has vertex (5, 3), but y = 2(x + 5)2 + 3 has vertex (-5, 3). The h is subtracted, so the sign flips.
Roots and the y-intercept
The y-intercept is c — set x = 0 and everything else disappears. No work at all.
The roots are where y = 0. Factor, or use the quadratic formula.
A parabola can have two roots, one (a repeated root, where the vertex sits on the axis), or none.
Symmetry gives free points
Once you know the vertex and one other point, its mirror image across the axis is also on the curve. Two calculations, five points.
Sketching it
Plot the vertex. Plot the y-intercept. Reflect that across the axis of symmetry for a third point.
Plot any roots you have. Then draw a smooth U through them — never a V, and never with straight sides.
Label the vertex and the axis. Those are what the marks are for.
The roots average to the axis
If the roots are 2 and 8, the axis is x = 5. Symmetry, and a quick check on your -b2a.
Maximum and minimum problems
"What is the greatest height?" asks for the y of the vertex. "When does it happen?" asks for the x.
For projectile problems, the roots are when the height is zero — launch and landing.
Read carefully which of the two the question wants; answering with the wrong coordinate is a common and avoidable loss.
Worked examples
Vertex and axis
For y = 2x2 - 12x + 7, find the axis of symmetry and the vertex.
- a = 2, b = -12 Read the coefficients, signs included.
- x = --122(2) = 124 = 3 The axis of symmetry.
- y = 2(9) - 12(3) + 7 = 18 - 36 + 7 = -11 Substitute back for the y.
- Vertex (3, -11), a minimum since a > 0. Opens upward.
Axis x = 3; vertex (3, -11), a minimum
Sketch from the key features
Sketch y = -x2 + 4x + 5.
- a = -1 < 0, so it opens downward. The vertex will be a maximum.
- Axis: x = -42(-1) = 2; y = -4 + 8 + 5 = 9 Vertex (2, 9).
- y-intercept is 5, at (0, 5). Its mirror is (4, 5). Both 2 units from the axis.
- Roots: -x2 + 4x + 5 = 0 gives (x - 5)(x + 1) = 0, so x = 5 and x = -1. Check: they average to 2 ✓
Vertex (2, 9), roots -1 and 5, y-intercept 5
A ball thrown from a roof
A ball is thrown straight up from the edge of a roof 80 feet high, at 64 feet per second. Gravity gives its height after t seconds as h(t) = -16t2 + 64t + 80 feet. How high does it go, when does it get there, and when does it hit the ground?
- a = -16 < 0, so the graph opens downward. Gravity pulls it back, so the vertex is the highest point.
- t = -642(-16) = 6432 = 2 seconds The time of the vertex.
- h(2) = -16(4) + 64(2) + 80 = -64 + 128 + 80 = 144 feet Substitute for the greatest height.
- The ground is h = 0: -16t2 + 64t + 80 = 0, so t2 - 4t - 5 = 0. Divide every term by -16.
- (t - 5)(t + 1) = 0, so t = 5 or t = -1. A time before the throw means nothing here — keep t = 5.
- Check: h(5) = -400 + 320 + 80 = 0 ✓ Up for 2 seconds, back to roof height at 4, the last 80 feet in one more.
Highest 144 feet, after 2 seconds; it lands after 5 seconds
Words to know
- 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 / zeros
- The x values where y = 0.
- Vertex form
- y = a(x - h)2 + k, with vertex at (h, k).
Mistakes to watch for
Dropping the minus in x = -b2a. With b = -12 it becomes --122a, which is positive. Writing the minus and the negative b both down stops the two from cancelling by accident.
Reading the vertex of a(x + 3)2 as x = 3. Vertex form is a(x - h)2 + k, so x + 3 means h = -3. Ask where the bracket equals zero and the sign sorts itself out.
Giving the time when the question asked for the height. The x of the vertex answers "when," the y answers "how high." Both are one substitution apart, and picking the wrong one is pure carelessness.
Drawing a parabola with straight sides or a sharp point. It curves everywhere and turns smoothly. A V belongs to absolute value, not to a quadratic.
A parabola reflects every incoming parallel ray to a single point. That is why satellite dishes, car headlights and radio telescopes are all parabolic.
Archimedes is said to have used bronze parabolic mirrors to set Roman ships alight during the siege of Syracuse. Modern tests suggest this is a legend.
But the underlying fact is real, and the word focus is Latin for hearth or fireplace — the point where the heat gathers.
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