8.3Vertex Form
Unlock the most powerful way to write a quadratic — vertex form y = a(x−h)²+k puts the vertex front and center, making transformations, max/min values, and real-world applications immediately readable.
Why This Matters
Vertex form makes it easy to identify the maximum or minimum of a quadratic — critical for optimization problems in business, engineering, and Physics. It's also the starting point for completing the square in Algebra 2.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does writing a quadratic in vertex form y = a(x−h)²+k make it easier to identify key features and describe transformations?
Learning Objectives
- 1Identify a, h, and k in vertex form y = a(x−h)²+k.
- 2State the vertex (h, k) and axis of symmetry x = h directly from vertex form.
- 3Determine whether the parabola opens up or down and whether the vertex is a max or min.
- 4Describe how a, h, and k each transform the parent function y = x².
- 5Graph a parabola given in vertex form using transformations.
- 6Convert from vertex form to standard form by expanding.
- 7Write vertex form given the vertex and one other point on the parabola.
- 8Interpret vertex form in real-world contexts (e.g., maximum height of a projectile).
Lesson Overview
Vertex form y = a(x−h)²+k is the most informative way to write a quadratic function. The vertex (h, k) is visible at a glance — no calculation needed. The value of a still controls direction and width, while h and k describe horizontal and vertical shifts from the parent parabola y = x². Mastering vertex form lets you graph quickly, solve real-world max/min problems, and convert between forms with confidence.
Worked Examples
Identify the vertex and axis of symmetry: y = 3(x − 4)² + 7
Vertex form: y = a(x−h)²+k
a=3, h=4, k=7
Vertex = (h, k) = (4, 7)
Axis of symmetry: x = h = 4
a=3>0 → opens upward → vertex is minimum
Identify the vertex: y = −2(x + 5)² − 3
Rewrite: y = −2(x − (−5))² + (−3)
h = −5, k = −3
Vertex = (−5, −3)
a=−2<0 → opens downward → vertex is maximum
Convert to standard form: y = 2(x − 3)² − 1
Expand (x−3)²: x²−6x+9
Distribute 2: 2x²−12x+18
Add −1: 2x²−12x+17
Convert to standard form: y = −(x + 2)² + 5
Rewrite: y = −1·(x+2)²+5
Expand (x+2)²: x²+4x+4
Distribute −1: −x²−4x−4
Add 5: −x²−4x+1
Write vertex form given vertex (2, −3) and point (4, 5)
Shell: y = a(x−2)²+(−3)
Substitute (4, 5): 5 = a(4−2)²−3
5 = a(4)−3 → 8 = 4a → a = 2
Vertex form: y = 2(x−2)²−3
Guided Practice
Guided Practice Video: Vertex Form of Quadratic Functions
Review vertex form y = a(x − h)² + k and how to identify the vertex, axis of symmetry, and direction of opening before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Identify the vertex and axis of symmetry: y = 5(x − 1)² + 9
Hint: Read h and k directly. Vertex = (h, k). Axis = x = h.
Identify the vertex: y = −(x + 6)² + 2
Hint: Rewrite as y = −(x − (−6))² + 2. What is h? What is k?
Does y = −4(x − 1)² + 8 open up or down? Is the vertex a max or min?
Hint: Look at the sign of a = −4.
Convert to standard form: y = (x − 5)² + 3
Hint: Expand (x−5)² first, then add 3.
Write vertex form: vertex (−1, 4), passes through (1, 12)
Hint: Shell: y = a(x+1)²+4. Substitute (1, 12) and solve for a.
Key Vocabulary
Vertex Form
y = a(x−h)²+k. The vertex (h, k) is read directly from the equation.
h (horizontal shift)
The x-coordinate of the vertex. Shifts the parabola right (h>0) or left (h<0). Watch the sign: (x−h) means h is positive.
k (vertical shift)
The y-coordinate of the vertex. Shifts the parabola up (k>0) or down (k<0).
Transformation
A change to the parent function y=x². Vertex form encodes three: vertical stretch/compression (a), horizontal shift (h), vertical shift (k).
Parent Function
The simplest form of a function family. For quadratics: y = x². Vertex at origin, opens upward, width = 1.
Completing the Square
An algebraic technique to rewrite a quadratic in vertex form by creating a perfect square trinomial.
Practice Questions
Interactive Practice — 5 Questions
What is the vertex of y = 2(x − 3)² + 5?
What is the vertex of y = −(x + 4)² − 2?
Convert y = (x − 1)² + 3 to standard form.
For y = 3(x − 2)² − 7, does the parabola open up or down?
Which vertex form equation has vertex (−1, 5)?
Independent Practice
Independent Practice
Identify the vertex and axis of symmetry: y = (x − 7)² + 2
Identify the vertex: y = −3(x + 1)² − 5
Does y = 2(x − 3)² − 4 open up or down? Max or min?
Convert to standard form: y = (x + 3)² − 2
Convert to standard form: y = −2(x − 1)² + 6
Common Mistakes
Reading the vertex as (h, k) from y = a(x + h)² + k — getting the sign of h wrong.
In y = a(x − h)² + k, the vertex is (h, k). If the equation shows (x + 3), then h = −3, not +3.
Confusing the direction of the vertical shift — thinking +k shifts down.
+k shifts the parabola up. −k shifts it down. The vertex moves to (h, k).
Forgetting that a negative value of a flips the parabola — assuming it always opens up.
If a < 0, the parabola opens downward and has a maximum. If a > 0, it opens upward and has a minimum.
Thinking |a| > 1 makes the parabola wider.
|a| > 1 makes the parabola narrower (steeper). |a| < 1 makes it wider (flatter).
Math Tips
In vertex form y = a(x − h)² + k, the vertex is always (h, k). Watch the sign: (x − 3)² has h = 3, not −3.
To convert to standard form, expand (x − h)² = x² − 2hx + h², then distribute a and add k.
The sign of a determines direction: a > 0 opens up (minimum), a < 0 opens down (maximum).
SAT tip: vertex form makes it easy to read the vertex directly — no formula needed.