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Algebra II

Quadratic Transformations

Lesson

Every parabola is a transformation of the parent function y=x2y = x^2. Once you can read the transformations, you can graph any quadratic from its equation.

Vertex form

y=a(x−h)2+ky = a(x - h)^2 + k
  • hh: horizontal shift (right if positive, left if negative). Vertex x is hh.
  • kk: vertical shift. Vertex y is kk.
  • aa: vertical stretch (∣a∣>1|a| > 1) or compression (∣a∣<1|a| < 1). Negative aa reflects over the x-axis.

Worked example 1 — read transformations

y=−2(x−3)2+5y = -2(x - 3)^2 + 5
  • Shifted 3 RIGHT (because −h=−3-h = -3)
  • Shifted 5 UP
  • Reflected over x-axis (negative aa)
  • Stretched by factor 2
  • Vertex: (3,5)(3, 5) — a MAX (opens down)

Convert standard → vertex with −b2a-\frac{b}{2a}

Given y=ax2+bx+cy = ax^2 + bx + c, the vertex x is h=−b2ah = -\frac{b}{2a}, then plug back in to find kk.

Worked example 2 — standard to vertex

y=x2+6x+8y = x^2 + 6x + 8

h=−6/2=−3h = -6 / 2 = -3. Plug in: (−3)2+6(−3)+8=9−18+8=−1(-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1.

Vertex form: y=(x+3)2−1y = (x + 3)^2 - 1, vertex (−3,−1)(-3, -1).

Reading the shape

  • Vertex form is the FASTEST read: vertex pops out, direction is the sign of aa.
  • a>0a > 0: opens UP, vertex is MIN. Range: y≥ky \ge k.
  • a<0a < 0: opens DOWN, vertex is MAX. Range: y≤ky \le k.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Vertex of y=(x−2)2+5\text{Vertex of } y = (x - 2)^2 + 5

Problem 2

Vertex of y=(x+4)2−3\text{Vertex of } y = (x + 4)^2 - 3

Problem 3

Vertex of y=x2+7\text{Vertex of } y = x^2 + 7

Problem 4

Stretch factor a of y=3x2\text{Stretch factor } a \text{ of } y = 3x^2

Practice

Standard problems matching the lesson.

Problem 5

Vertex of y=−(x−1)2+6\text{Vertex of } y = -(x - 1)^2 + 6

Problem 6

Vertical shift (up) of y=x2+8 from parent y=x2\text{Vertical shift (up) of } y = x^2 + 8 \text{ from parent } y = x^2

Problem 7

Horizontal shift right of y=(x−5)2 from parent\text{Horizontal shift right of } y = (x - 5)^2 \text{ from parent}

Problem 8

Stretch factor a of y=2(x−3)2+1\text{Stretch factor } a \text{ of } y = 2(x - 3)^2 + 1

Problem 9

y = -4(x+1)^2 - 2 opens up (1) or down (0)?

Problem 10

y-intercept of y=(x−2)2+1\text{y-intercept of } y = (x - 2)^2 + 1

Problem 11

y-intercept of y=2(x−1)2+3\text{y-intercept of } y = 2(x - 1)^2 + 3

Problem 12

Value of y=(x−1)2+4 at x=2\text{Value of } y = (x - 1)^2 + 4 \text{ at } x = 2

Problem 13

Value of y=−(x−1)2+10 at x=3\text{Value of } y = -(x - 1)^2 + 10 \text{ at } x = 3

Problem 14

Vertex of y=(x−6)2−4\text{Vertex of } y = (x - 6)^2 - 4

Problem 15

Vertex of y=3(x+2)2−9\text{Vertex of } y = 3(x + 2)^2 - 9

Problem 16

y=(x−h)2 is y=x2 shifted 3 right.h=?\begin{gathered}y = (x - h)^2 \text{ is } y = x^2 \text{ shifted 3 right.} \\ h = ?\end{gathered}

Problem 17

h(t) = -(t-3)^2 + 12. Vertex?

Problem 18

Min y of y = 2(x-1)^2 - 5

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Vertex of y=−2(x−4)2+1\text{Vertex of } y = -2(x - 4)^2 + 1

Problem 20

Vertex of y=x2+6x+8\text{Vertex of } y = x^2 + 6x + 8

Problem 21

Vertex of y=x2−8x+17\text{Vertex of } y = x^2 - 8x + 17

Problem 22

Stretch factor a of y=−12(x−1)2\text{Stretch factor } a \text{ of } y = -\tfrac{1}{2}(x - 1)^2

Problem 23

y-intercept of y=−(x+3)2+5\text{y-intercept of } y = -(x + 3)^2 + 5

Problem 24

Value of y=2(x−3)2−7 at x=−1\text{Value of } y = 2(x - 3)^2 - 7 \text{ at } x = -1

Problem 25

y at x=5 of y = 3(x-2)^2

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