Algebra 1 · Unit 8
Find zeros, y-intercept, vertex, domain, and range from an equation or graph, and interpret them in a story.
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Vertex and intercepts
The y-intercept is f(0). The zeros are where f(x) = 0, the x-intercepts. The vertex is the turning point: a minimum if a > 0, a maximum if a < 0.
Axis of symmetry
x = −b2a. The vertex has that x-value, and y = f of that x.
Domain and range
On the coordinate plane, domain is all real numbers. Range is y ≥ k if the parabola opens up, or y ≤ k if it opens down. A story can cut the domain.
In a story
The vertex is a highest or lowest value (max height, min cost). A zero can be when something hits the ground or breaks even.
Read a, the y-intercept, or a vertex from vertex form.
a = 3 > 0, so it opens up.
f(0) = 5
The y-intercept is (0, 5).
Answer: opens up; (0, 5)
Vertex form a(x − h)² + k
Vertex (2, 5)
Answer: (2, 5)
a = −2 < 0, so it opens down.
The vertex is a maximum.
Answer: opens down; maximum
f(0) = 8
(0, 8)
Answer: (0, 8)
x = −b2a
x = −(−8)2(1)
x = 4
Answer: x = 4
h = −2, k = 3
Vertex (−2, 3). Opens down.
Answer: (−2, 3)
Zeros, vertex from standard form, and a story.
x = −(−4)2(1) = 2
f(2) = 4 − 8 + 1 = −3
Vertex (2, −3)
Answer: (2, −3)
(x − 2)(x − 4) = 0
Zeros (2, 0) and (4, 0)
Answer: (2, 0) and (4, 0)
Domain: all real numbers.
Vertex (1, 4), opens up.
Range: y ≥ 4
Answer: all reals; y ≥ 4
t = −482(−16)
t = −48−32
t = 32
h(32) = −16(94) + 4832 + 4
= −36 + 72 + 4 = 40
Maximum 40 feet after 32 seconds.
Answer: 32 s; 40 ft
Zeros x = −1 and x = 5
f(0) = (1)(−5) = −5
y-intercept (0, −5)
Answer: (−1, 0), (5, 0), (0, −5)
n = −202(−2) = 5
P(5) = −2(25) + 100 − 32 = 18
5 items; 18 dollars.
Answer: 5 items; 18 dollars
Restricted domain, or a vertex that needs more work.
Domain in the story: 0 ≤ t ≤ 4
t = −642(−16) = 2
h(2) = −16(4) + 128 + 5 = 69
h(0) = 5 and h(4) = 5
The vertex is on the interval, so the range is 5 ≤ h ≤ 69.
Answer: t in [0, 4]; vertex (2, 69); 5 ≤ h ≤ 69
x = −82(2) = −2
f(−2) = 2(4) − 16 − 1 = −9
Vertex (−2, −9)
Answer: (−2, −9)
A(x) = 20x − x²
x = −202(−1) = 10
A(10) = 100
x = 0 is a side of length 0, so the garden vanishes. Not a useful garden, but it is a zero of A.
Answer: 100 sq ft at x = 10
Opens down, vertex (3, 7) is in the interval, so the max is 7.
f(1) = −1(4) + 7 = 3
f(6) = −1(9) + 7 = −2
The smaller endpoint is −2, so the range is −2 ≤ y ≤ 7.
Answer: −2 ≤ y ≤ 7
2(x² − x − 6) = 0
2(x − 3)(x + 2) = 0
x = 3 or x = −2
Answer: x = −2, x = 3
Vertex t = −302(−5) = 3
h(3) = −5(9) + 90 + 10 = 55
Yes. It reaches 55 m exactly at the vertex.
Answer: yes, at t = 3 s
Axis halfway between 1 and 5: x = 3
f(x) = a(x − 1)(x − 5)
−10 = a(0 − 1)(0 − 5)
−10 = 5a
a = −2
f(3) = −2(2)(−2) = 8
Vertex (3, 8). a < 0, so a maximum.
Answer: (3, 8); maximum
The formula eventually goes below 0 after it hits the ground.
Height in the story stops at 0. Domain in the story stops at the positive zero.
Answer: height cannot be negative
Axis x = −42a = −2a
The vertex y-value is 0, so f of that x is 0.
Discriminant 16 − 4a(1) = 0 for a double zero at the vertex.
16 − 4a = 0
a = 4
x = −24 = −12
Vertex (−12, 0)
Answer: a = 4; (−1/2, 0)
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