Quadratic Vertex Form

Grade 10 · mathematics · 100 practice problems · read aloud

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Quadratic Vertex Form: Your Graphing Superpower 🚀

What is Vertex Form?

The vertex form of a quadratic equation is: y = a(x - h)² + k. This form is incredibly useful because it immediately shows you the parabola's vertex (its highest or lowest point) at (h, k). This makes graphing the parabola quick and easy!

Step-by-Step Guide

  1. Identify the Vertex: The vertex is the point (h, k). Watch the sign! In y = a(x - h)² + k, 'h' in the equation is the opposite of the x-coordinate.
  2. Determine Direction: Look at the value of 'a'. If a > 0, the parabola opens upwards (U-shaped). If a < 0, it opens downwards (n-shaped).
  3. Plot Key Points: Start at the vertex. Use the value of 'a' to find other points. For example, from the vertex, move right 1 unit, then up 'a' units to find the next point.

Visual Examples

Example 1: Graph y = 2(x - 1)² - 3

  • Vertex (h, k): (1, -3)
  • Direction (a): a = 2 (positive, so opens UP)
  • Plot: Start at (1, -3). Since a=2, from the vertex, go right 1, up 2 to (2, -1). This pattern helps sketch the curve.

Example 2: Find the vertex of y = -½(x + 4)² + 5

  • Rewrite as y = -½(x - (-4))² + 5
  • Vertex (h, k): (-4, 5)
  • Direction (a): a = -½ (negative, so opens DOWN)

🚨 Common Mistakes to Avoid

  • Sign Error with h: The biggest mistake! In y = a(x - h)² + k, the x-coordinate is h, not -h. For (x + 4)², it's (x - (-4))², so h = -4.
  • Ignoring 'a': Forgetting that 'a' affects the width and direction of the parabola. If |a| > 1, the parabola is skinnier. If |a| < 1, it's wider.
  • Confusing Vertex and y-intercept: The vertex is (h, k). The y-intercept is found by setting x = 0 and solving for y.

💡 Tips & Tricks

  • Memory Aid: For the vertex (h, k), remember "Hey (h), okay (k)?" and that the sign of 'h' inside the parentheses is the opposite of what you see.
  • Graphing Shortcut: From the vertex, use the pattern "over 1, up a; over 2, up 4a" to plot symmetric points quickly.
  • Check Your Work: The axis of symmetry is always the vertical line x = h.

Practice Suggestions

To master vertex form:

  1. Start by identifying the vertex and direction for 10 different equations.
  2. Practice converting from standard form (ax² + bx + c) to vertex form by completing the square.
  3. Create your own equations in vertex form and graph them by hand, labeling the vertex and axis of symmetry.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 Convert f(x) = 2x² - 12x + 19 to vertex form = ?

Hint: Complete the square by factoring out the leading coefficient and finding the value that creates a perfect square trinomial.

Show the answer

Answer: 2(x - 3)² + 1

  1. Factor out the coefficient of x² from the first two terms** f(x) = 2x² - 12x + 19 Factor 2 from 2x² - 12x: f(x) = 2(x² - 6x) + 19 --- **
  2. Complete the square inside the parentheses** Inside the parentheses: x² - 6x Take half of the coefficient of x: (-6)/2 = -3 Square it: (-3)² = 9 Add and subtract 9 inside the parentheses: f(x) = 2(x² - 6x + 9 - 9) + 19 --- **
  3. Rewrite the perfect square trinomial and simplify** x² - 6x + 9 is (x - 3)² So: f(x) = 2[(x - 3)² - 9] + 19 --- **
  4. Distribute the 2 and combine constants** f(x) = 2(x - 3)² - 18 + 19 f(x) = 2(x - 3)² + 1 --- **Final answer:** 2(x - 3)² + 1

Let's convert f(x) = 2x² - 12x + 19 to vertex form step-by-step. Vertex form is: a(x - h)² + k --- **

2 Convert f(x) = 2x² - 12x + 16 to vertex form = ?

Hint: To convert a quadratic from standard form to vertex form, complete the square by factoring out the leading coefficient and finding the value that creates a perfect square trinomial.

Show the answer

Answer: 2(x - 3)² - 2

  1. Factor out the coefficient of x² from the first two terms f(x) = 2(x² - 6x) + 16
  2. Complete the square inside the parentheses - Look at the x term: -6x - Take half of -6: -3 - Square it: (-3)² = 9 - Add and subtract 9 inside the parentheses: f(x) = 2(x² - 6x + 9 - 9) + 16
  3. Rewrite the expression - The first three terms form a perfect square: x² - 6x + 9 = (x - 3)² - Keep the -9 inside: f(x) = 2[(x - 3)² - 9] + 16
  4. Distribute the 2 f(x) = 2(x - 3)² - 18 + 16
  5. Simplify the constants f(x) = 2(x - 3)² - 2 This is the vertex form: 2(x - 3)² - 2 The vertex is at (3, -2), and the parabola opens upward since the coefficient 2 is positive.

Let's convert f(x) = 2x² - 12x + 16 to vertex form step-by-step. Vertex form is: a(x - h)² + k

3 Convert f(x) = 3x² - 18x + 25 to vertex form = ?

Hint: Complete the square by factoring out the leading coefficient and finding the value that creates a perfect square trinomial.

Show the answer

Answer: 3(x - 3)² - 2

  1. Factor out the coefficient of x² from the first two terms: f(x) = 3(x² - 6x) + 25
  2. Complete the square inside the parentheses: Take half of -6, which is -3, and square it to get 9
  3. Add and subtract 9 inside the parentheses: f(x) = 3(x² - 6x + 9 - 9) + 25
  4. Rewrite as: f(x) = 3[(x² - 6x + 9) - 9] + 25
  5. Simplify: f(x) = 3[(x - 3)² - 9] + 25
  6. Distribute the 3: f(x) = 3(x - 3)² - 27 + 25
  7. Combine constants: f(x) = 3(x - 3)² - 2 The vertex form is 3(x - 3)² - 2.

4 y = 7(x - 8)² + 9. Find vertex and axis of symmetry.

Hint: Recall that in vertex form y = a(x - h)² + k, the vertex is (h, k) and the axis of symmetry is the vertical line through the vertex.

Show the answer

Answer: Vertex: (8, 9), Axis of symmetry: x = 8

  1. Identify h and k from the vertex form y = a(x - h)² + k. Here, y = 7(x - 8)² + 9, so h = 8 and k = 9.
  2. The vertex is (h, k) = (8, 9).
  3. The axis of symmetry is the vertical line x = h, so x = 8. Final answer: Vertex: (8, 9), Axis of symmetry: x = 8.

5 y = 3(x - 5)^2 - 7. Find vertex and axis of symmetry.

Hint: In vertex form y = a(x - h)^2 + k, the vertex is at (h, k) and the axis of symmetry is the vertical line through the vertex.

Show the answer

Answer: Vertex: (5, -7), Axis: x = 5

  1. Identify h and k from the equation y = 3(x - 5)^2 - 7. Here, h = 5 and k = -7.
  2. The vertex is (h, k) = (5, -7).
  3. The axis of symmetry is the vertical line x = h, so x = 5. Final answer: Vertex: (5, -7), Axis: x = 5.

6 y = 4(x - 2)^2 - 8. Find vertex and axis of symmetry.

Hint: Recall that in vertex form y = a(x - h)^2 + k, the vertex is at (h, k) and the axis of symmetry is the vertical line x = h. Identify h and k from the given equation.

Show the answer

Answer: Vertex: (2, -8), Axis: x = 2

  1. The given equation is y = 4(x - 2)^2 - 8, which is in vertex form y = a(x - h)^2 + k.
  2. Compare: (x - h) = (x - 2), so h = 2. The constant term is k = -8.
  3. Therefore, the vertex is (h, k) = (2, -8).
  4. The axis of symmetry is the vertical line through the vertex, x = h, so x = 2.

The answer is Vertex: (2, -8), Axis: x = 2.

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