IB AA · 3B Sketching Graphs
Find the turning point, then read the range from it.
Axis
For y = ax² + bx + c, the line of symmetry is x = −b2a.
Vertex
Substitute that x into the function. The point is the vertex.
Range
If a > 0, y is at least the vertex height. If a < 0, y is at most that height.
A log of a quadratic
Complete the square inside the log. The vertex of the inside, when the inside stays positive, is the vertex of the log graph.
a is 1 or −1.
f(x) = x² − 4x + 1
x = −b2a
x = 42
x = 2
f(2) = (2)² − 4(2) + 1
f(2) = −3
a is positive, so the smallest output is −3.
The range is y ≥ −3.
The vertex is (2, −3).
The line of symmetry is x = 2.
Answer: vertex (2, −3), symmetry x = 2, y ≥ −3
f(x) = x² + 2x + 3
x = −b2a
x = −22
x = −1
f(−1) = (−1)² + 2(−1) + 3
f(−1) = 2
a is positive, so the smallest output is 2.
The range is y ≥ 2.
The vertex is (−1, 2).
The line of symmetry is x = −1.
Answer: vertex (−1, 2), symmetry x = −1, y ≥ 2
f(x) = −x² + 5
x = −b2a
x = 0−2
x = 0
f(0) = −(0)² + 5
f(0) = 5
a is negative, so the largest output is 5.
The range is y ≤ 5.
The vertex is (0, 5).
The line of symmetry is x = 0.
Answer: vertex (0, 5), symmetry x = 0, y ≤ 5
f(x) = x² − 8x + 10
x = −b2a
x = 82
x = 4
f(4) = (4)² − 8(4) + 10
f(4) = −6
a is positive, so the smallest output is −6.
The range is y ≥ −6.
The vertex is (4, −6).
The line of symmetry is x = 4.
Answer: vertex (4, −6), symmetry x = 4, y ≥ −6
f(x) = −x² + 4x + 1
x = −b2a
x = −4−2
x = 2
f(2) = −(2)² + 4(2) + 1
f(2) = 5
a is negative, so the largest output is 5.
The range is y ≤ 5.
The vertex is (2, 5).
The line of symmetry is x = 2.
Answer: vertex (2, 5), symmetry x = 2, y ≤ 5
f(x) = 2x² − 12x + 7
x = −b2a
x = 124
x = 3
f(3) = 2(3)² − 12(3) + 7
f(3) = −11
a is positive, so the smallest output is −11.
The range is y ≥ −11.
The vertex is (3, −11).
The line of symmetry is x = 3.
Answer: vertex (3, −11), symmetry x = 3, y ≥ −11
a is not 1.
f(x) = x² + 6x + 4
x = −b2a
x = −62
x = −3
f(−3) = (−3)² + 6(−3) + 4
f(−3) = −5
a is positive, so the smallest output is −5.
The range is y ≥ −5.
The vertex is (−3, −5).
The line of symmetry is x = −3.
Answer: vertex (−3, −5), symmetry x = −3, y ≥ −5
f(x) = −2x² + 8x + 1
x = −b2a
x = −8−4
x = 2
f(2) = −2(2)² + 8(2) + 1
f(2) = 9
a is negative, so the largest output is 9.
The range is y ≤ 9.
The vertex is (2, 9).
The line of symmetry is x = 2.
Answer: vertex (2, 9), symmetry x = 2, y ≤ 9
f(x) = x² − 6
x = −b2a
x = 02
x = 0
f(0) = (0)² − 6
f(0) = −6
a is positive, so the smallest output is −6.
The range is y ≥ −6.
The vertex is (0, −6).
The line of symmetry is x = 0.
Answer: vertex (0, −6), symmetry x = 0, y ≥ −6
f(x) = x² − 2x − 3
x = −b2a
x = 22
x = 1
f(1) = (1)² − 2(1) − 3
f(1) = −4
a is positive, so the smallest output is −4.
The range is y ≥ −4.
The vertex is (1, −4).
The line of symmetry is x = 1.
Answer: vertex (1, −4), symmetry x = 1, y ≥ −4
f(x) = 3x² − 6x + 4
x = −b2a
x = 66
x = 1
f(1) = 3(1)² − 6(1) + 4
f(1) = 1
a is positive, so the smallest output is 1.
The range is y ≥ 1.
The vertex is (1, 1).
The line of symmetry is x = 1.
Answer: vertex (1, 1), symmetry x = 1, y ≥ 1
f(x) = −x² − 2x + 8
x = −b2a
x = 2−2
x = −1
f(−1) = −(−1)² − 2(−1) + 8
f(−1) = 9
a is negative, so the largest output is 9.
The range is y ≤ 9.
The vertex is (−1, 9).
The line of symmetry is x = −1.
Answer: vertex (−1, 9), symmetry x = −1, y ≤ 9
A log, a reciprocal, and a quadratic with a fraction vertex.
y = ln(x² − 6x + 13)
x² − 6x + 13 = (x − 3)² + 4
The inside is smallest when x = 3, and that smallest value is 4.
The vertex is (3, ln(4)).
The line of symmetry is x = 3.
Answer: vertex (3, ln(4)), symmetry x = 3
x² − 6x + 13 = (x − 3)² + 4
The denominator is symmetric about x = 3, and its smallest value is 4.
The line of symmetry is x = 3.
The largest value of y is 14.
The denominator is always positive, so y is positive and never passes that largest value.
The range is 0 < y ≤ 14.
Answer: x = 3, 0 < y ≤ 14
f(x) = −2x² + 7
x = −b2a
x = 0−4
x = 0
f(0) = −2(0)² + 7
f(0) = 7
a is negative, so the largest output is 7.
The range is y ≤ 7.
The vertex is (0, 7).
The line of symmetry is x = 0.
Answer: vertex (0, 7), symmetry x = 0, y ≤ 7
f(x) = x² − 10x + 21
x = −b2a
x = 102
x = 5
f(5) = (5)² − 10(5) + 21
f(5) = −4
a is positive, so the smallest output is −4.
The range is y ≥ −4.
The vertex is (5, −4).
The line of symmetry is x = 5.
Answer: vertex (5, −4), symmetry x = 5, y ≥ −4
f(x) = x² − 6x + 2
x = −b2a
x = 62
x = 3
f(3) = (3)² − 6(3) + 2
f(3) = −7
a is positive, so the smallest output is −7.
The range is y ≥ −7.
The vertex is (3, −7).
The line of symmetry is x = 3.
Answer: vertex (3, −7), symmetry x = 3, y ≥ −7
f(x) = −x² + 6x − 4
x = −b2a
x = −6−2
x = 3
f(3) = −(3)² + 6(3) − 4
f(3) = 5
a is negative, so the largest output is 5.
The range is y ≤ 5.
The vertex is (3, 5).
The line of symmetry is x = 3.
Answer: vertex (3, 5), symmetry x = 3, y ≤ 5
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