Quadratic Functions

Grade 9 · algebra · 38 practice problems · read aloud

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Quadratic Functions: The U-Shaped Graphs

A quadratic function creates a smooth, U-shaped curve called a parabola. It's written as f(x) = ax² + bx + c. You see these everywhere—in the arc of a basketball, the design of a satellite dish, and the support cables of a bridge! Understanding them helps us model and predict real-world motion and shapes.

Key Parts of a Parabola

  • Vertex: The highest or lowest point of the parabola.
  • Axis of Symmetry: The vertical line that folds the parabola perfectly in half. It goes through the vertex.
  • Roots (or Zeros): The x-values where the parabola crosses the x-axis (where y=0).

How to Graph a Quadratic Function

  1. Identify a, b, and c from f(x) = ax² + bx + c.
  2. Find the Vertex. The x-coordinate is x = -b/(2a). Plug this back into the function to find the y-coordinate.
  3. Find the Y-Intercept. This is the point (0, c). It's where the graph crosses the y-axis.
  4. Plot Points. Use the axis of symmetry to find points on the other side of the vertex. Connect them to form the U-shape!

Example 1: Graphing f(x) = x² - 4x + 3

  1. Identify: a=1, b=-4, c=3.
  2. Vertex: x = -(-4)/(2*1) = 4/2 = 2. f(2) = (2)² - 4(2) + 3 = -1. Vertex is (2, -1).
  3. Y-Intercept: (0, 3).
  4. Plot: The axis of symmetry is x=2. The point (0,3) is 2 units left of the axis, so plot (4,3) 2 units right. Connect the points!

Example 2: Finding Roots of f(x) = 2x² + 4x - 6

Set the function equal to zero and solve: 2x² + 4x - 6 = 0

  1. Divide by 2: x² + 2x - 3 = 0
  2. Factor: (x + 3)(x - 1) = 0
  3. Solve: x = -3 and x = 1. The roots are -3 and 1.

🚨 Common Mistakes to Avoid

  • Wrong Vertex Formula: The x-coordinate is -b/(2a), not -b/(a) or b/(2a). Always use the negative sign!
  • Sign Errors: Be super careful with positive and negative values for b and c when calculating.
  • Direction of Opening: If a is positive, the parabola opens UP (U-shaped). If a is negative, it opens DOWN (n-shaped).

💡 Tips & Tricks

  • Vertex Shortcut: Once you find the vertex's x-value, the Axis of Symmetry is the vertical line at that x (e.g., x=2).
  • Memory Aid: For the vertex formula, remember "Negative B, over 2A" like a little song.
  • Check Your Work: The y-intercept (0, c) is a quick and easy point to plot first.

How to Practice

Start simple! Graph functions like f(x) = x² + 2 and f(x) = -x² + 4. Identify the vertex, axis of symmetry, and y-intercept for each. Then, move to functions that need factoring to find the roots. Use graph paper or a digital graphing tool to check your sketches.

Practice problems

6 of the 38, worked through step by step — try them before opening the answer.

1 Convert y = 3(x - 7)² + 9 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient and combine like terms

Show the answer

Answer: y = 3x² - 42x + 156

  1. Start with y = 3(x - 7)² + 9
  2. Expand (x - 7)² = x² - 14x + 49
  3. Multiply by 3: 3(x² - 14x + 49) = 3x² - 42x + 147
  4. Add the constant term: 3x² - 42x + 147 + 9
  5. Combine like terms: 3x² - 42x + 156 The final answer is y = 3x² - 42x + 156

2 Convert y = 2(x - 1)² + 6 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient and combine like terms

Show the answer

Answer: y = 2x² - 4x + 8

  1. Start with the vertex form: y = 2(x - 1)² + 6
  2. Expand (x - 1)² = (x - 1)(x - 1) = x² - 2x + 1
  3. Multiply by the coefficient 2: 2(x² - 2x + 1) = 2x² - 4x + 2
  4. Add the constant term: 2x² - 4x + 2 + 6 = 2x² - 4x + 8
  5. The standard form is y = 2x² - 4x + 8

3 Convert y = 2(x - 7)² + 9 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient, and finally combine like terms

Show the answer

Answer: y = 2x² - 28x + 107

  1. Start with y = 2(x - 7)² + 9
  2. Expand (x - 7)² = x² - 14x + 49
  3. Multiply by 2: 2(x² - 14x + 49) = 2x² - 28x + 98
  4. Add the constant term: 2x² - 28x + 98 + 9
  5. Combine like terms: 2x² - 28x + 107

The answer is y = 2x² - 28x + 107.

4 Convert y = 3(x - 5)² + 7 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient and combine like terms

Show the answer

Answer: y = 3x² - 30x + 82

  1. Start with y = 3(x - 5)² + 7
  2. Expand (x - 5)² = x² - 10x + 25
  3. Multiply by 3: 3(x² - 10x + 25) = 3x² - 30x + 75
  4. Add the constant term: 3x² - 30x + 75 + 7
  5. Combine like terms: 3x² - 30x + 82

The answer is y = 3x² - 30x + 82.

5 Convert y = 3(x - 8)² + 9 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient and combine like terms

Show the answer

Answer: y = 3x² - 48x + 201

  1. Start with y = 3(x - 8)² + 9
  2. Expand (x - 8)² = x² - 16x + 64
  3. Multiply by 3: 3(x² - 16x + 64) = 3x² - 48x + 192
  4. Add the constant term: 3x² - 48x + 192 + 9
  5. Combine like terms: 3x² - 48x + 201 Final answer: y = 3x² - 48x + 201

6 Convert y = 4(x - 2)² + 6 to standard form

Hint: Expand the squared binomial first, then distribute the coefficient, and finally combine like terms

Show the answer

Answer: y = 4x² - 16x + 22

  1. Start with y = 4(x - 2)² + 6
  2. Expand (x - 2)² = x² - 4x + 4
  3. Multiply by 4: 4(x² - 4x + 4) = 4x² - 16x + 16
  4. Add the constant term: 4x² - 16x + 16 + 6
  5. Combine like terms: 4x² - 16x + 22 The final answer is y = 4x² - 16x + 22
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