Graph Key Features

Grade 9 · mathematics · 87 practice problems · read aloud

🔊 Listen to this explanation

📈 Graph Key Features

Graph key features are the specific, important points and characteristics of a graph that tell us its story. Identifying them helps us understand the behavior of functions and relationships between variables, which is essential in algebra and beyond.

Step-by-Step Guide

  1. Find the Y-Intercept: Where does the graph cross the y-axis? Set x=0 and solve for y.
  2. Find the X-Intercepts (Zeros/Roots): Where does the graph cross the x-axis? Set y=0 and solve for x.
  3. Find the Vertex: For a parabola, this is the highest or lowest point. Use the formula x = -b/(2a) for a quadratic in the form y=ax²+bx+c.
  4. Determine the Axis of Symmetry: The vertical line that splits the parabola into mirror images (it goes through the vertex!).
  5. Check the Direction: Does the parabola open upwards (like a 'U') or downwards (like an 'n')? Look at the sign of the 'a' coefficient.

Visual Examples

Example 1: y = x² - 4

  1. Y-Intercept: Set x=0. y = (0)² - 4 = -4. Point: (0, -4)
  2. X-Intercepts: Set y=0. 0 = x² - 4 → x²=4 → x=2, -2. Points: (2, 0) and (-2, 0)
  3. Vertex: a=1, b=0. x = -0/(2*1) = 0. Plug in x=0: y=-4. Vertex: (0, -4)
  4. Axis of Symmetry: x = 0 (the y-axis!)
  5. Direction: a=1 (positive), so it opens upwards.

Example 2: y = -x² + 2x + 3

  1. Y-Intercept: (0, 3)
  2. X-Intercepts: 0 = -x² + 2x + 3 → 0 = -(x² - 2x - 3) → 0 = -(x-3)(x+1). So, x=3 and x=-1. Points: (3, 0) and (-1, 0)
  3. Vertex: x = -2/(2*(-1)) = 1. Plug in x=1: y = -(1)² + 2(1) + 3 = 4. Vertex: (1, 4)
  4. Axis of Symmetry: x = 1
  5. Direction: a=-1 (negative), so it opens downwards.

⚠️ Common Mistakes

  • Mixing up x and y-intercepts: Remember, the x-intercept has a y-value of 0. The y-intercept has an x-value of 0.
  • Vertex Formula Error: The formula is x = -b/(2a). Forgetting the negative sign is a common error.
  • Direction Confusion: If 'a' is positive, the graph opens UP. If 'a' is negative, it opens DOWN. Think: "Positive is happy (U), negative is sad (n)."

💡 Tips & Tricks

  • ROXS: Remember "Roots (x-intercepts) are on the X-axis." This helps you remember which coordinate is zero.
  • Sketch it! Even a rough sketch after finding a few points can help you visualize all the key features and check your work.
  • Vertex Shortcut: The x-coordinate of the vertex is always the average of the two x-intercepts! (For Example 2: (-1 + 3)/2 = 1).

Practice Suggestions

Start simple! Grab a quadratic equation, list out the steps, and find all five key features. Use online graphing tools like Desmos to check your answers instantly. Try creating your own equations with specific intercepts or vertices and see if you can graph them correctly.

Practice problems

6 of the 87, worked through step by step — try them before opening the answer.

1 f(x) = x² - 4x + 3; f(2) = ?

Hint: To evaluate a function at a specific value, substitute the given number for the variable in the function's expression and simplify using order of operations.

Show the answer

Answer: -1

  1. Write the function with x replaced by 2. f(2) = (2)² - 4*(2) + 3
  2. Calculate (2)². (2)² = 4 So now: f(2) = 4 - 4*(2) + 3
  3. Calculate 4*(2). 4*(2) = 8 So now: f(2) = 4 - 8 + 3
  4. Perform the subtraction and addition from left to right. 4 - 8 = -4 Then: -4 + 3 = -1 So the final answer is: f(2) = -1

We are given the function: f(x) = x² - 4x + 3 We want to find f(2), which means we substitute x = 2 into the function.

2 f(x) = 2x² - 8x + 3; f(5) = ?

Hint: Substitute the given x-value into the function and follow the order of operations.

Show the answer

Answer: 13

  1. Start with the function f(x) = 2x² - 8x + 3
  2. Substitute x = 5 into the function: f(5) = 2(5)² - 8(5) + 3
  3. Calculate the exponent first: (5)² = 25
  4. Multiply: 2 × 25 = 50 and -8 × 5 = -40
  5. Combine all terms: 50 - 40 + 3
  6. Simplify: 50 - 40 = 10, then 10 + 3 = 13

The answer is 13.

3 f(x) = 2x² - 5x + 1; f(3) = ?

Hint: Substitute the given x-value into the function and follow the order of operations.

Show the answer

Answer: 4

  1. Write the function: f(x) = 2x² - 5x + 1
  2. Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 1
  3. Calculate the exponent first: 3² = 9
  4. Multiply: 2 × 9 = 18 and 5 × 3 = 15
  5. Rewrite the expression: f(3) = 18 - 15 + 1
  6. Perform addition and subtraction from left to right: 18 - 15 = 3, then 3 + 1 = 4

The answer is 4.

4 f(x) = 3x² - 12x + 5; f(4) = ?

Hint: Substitute the given x-value into the function and follow the order of operations.

Show the answer

Answer: 5

  1. Write the function: f(x) = 3x² - 12x + 5
  2. Substitute x = 4: f(4) = 3(4)² - 12(4) + 5
  3. Calculate the exponent: (4)² = 16
  4. Multiply: 3 × 16 = 48 and 12 × 4 = 48
  5. Substitute back: f(4) = 48 - 48 + 5
  6. Simplify: 48 - 48 = 0, then 0 + 5 = 5

The answer is 5.

5 f(x) = 3x² - 12x + 7; f(4) = ?

Hint: Substitute the given x-value into the function and follow the order of operations carefully.

Show the answer

Answer: 7

  1. Write the function: f(x) = 3x² - 12x + 7
  2. Substitute x = 4 into the function: f(4) = 3(4)² - 12(4) + 7
  3. Calculate the exponent first: (4)² = 16
  4. Multiply: 3 × 16 = 48
  5. Multiply: 12 × 4 = 48
  6. Rewrite the expression: f(4) = 48 - 48 + 7
  7. Perform subtraction: 48 - 48 = 0
  8. Add the remaining term: 0 + 7 = 7

The answer is 7.

6 f(x) = 2x² - 8x + 6, vertex = ?

Hint: For a quadratic function in standard form, the x-coordinate of the vertex can be found using a specific formula involving the coefficients. Once you have the x-value, substitute it back into the function to find the y-coordinate.

Show the answer

Answer: (2, -2)

  1. Identify the coefficients a, b, and c from the function. a = 2 b = -8 c = 6
  2. Calculate the x-coordinate of the vertex using the formula. x = -b / (2a) x = -(-8) / (2 * 2) x = 8 / 4 x = 2
  3. Calculate the y-coordinate of the vertex by substituting x = 2 into the original function. f(2) = 2*(2)² - 8*(2) + 6 f(2) = 2*4 - 16 + 6 f(2) = 8 - 16 + 6 f(2) = -8 + 6 f(2) = -2
  4. Write the vertex as an ordered pair (x, y). The vertex is (2, -2). Therefore, the vertex of the function f(x) = 2x² - 8x + 6 is (2, -2).

To find the vertex of the quadratic function f(x) = 2x² - 8x + 6, we can use the vertex formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by: x = -b / (2a)

Practise this topic — 10 free problems, no signup →