Construct Functions

Grade 8 · algebra · 100 practice problems · read aloud

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Constructing Functions

🤔 What is it and Why is it Useful?

Constructing a function means writing an equation (usually y = mx + b) that describes a relationship between two quantities. It's like finding the secret rule that turns an input (x) into an output (y). This is super useful for making predictions! For example, you can write a function to calculate the total cost of video games or how far a car travels over time.

📝 Step-by-Step Guide

  1. Identify the variables: Figure out which quantity depends on the other. The dependent variable is y (the output), and the independent variable is x (the input).
  2. Find the rate of change (slope, m): How much does y change when x increases by 1? Use the formula: m = (change in y) / (change in x).
  3. Find the starting value (y-intercept, b): What is the value of y when x is 0? This is often the initial value or flat fee.
  4. Write the equation: Plug your values for m and b into the equation y = mx + b.

🔢 Visual Examples

Example 1: The Taxi Fare

A taxi charges a $3 flat fee plus $2 per mile.

  1. Variables: x = miles, y = total cost.
  2. Slope (m): The cost per mile is $2. So, m = 2.
  3. Y-intercept (b): The flat fee when miles (x) is 0 is $3. So, b = 3.
  4. Equation: y = 2x + 3

Example 2: From a Table

Find the function for this table:

xy
05
18
211
  1. Slope (m): As x increases by 1, y increases by 3. So, m = 3.
  2. Y-intercept (b): When x=0, y=5. So, b = 5.
  3. Equation: y = 3x + 5

⚠️ Common Mistakes

  • Mixing up x and y: Remember, x is the input (cause) and y is the output (effect).
  • Wrong slope sign: If y decreases as x increases, the slope (m) is negative!
  • Forgetting the y-intercept: If the pattern doesn't start at (0, b), you might need to calculate b using the equation after finding m.

💡 Tips & Tricks

  • Look for keywords: "per" often indicates the slope (m), and "initial value" or "flat fee" often indicates the y-intercept (b).
  • Use the phrase "My Brother Eats Burgers" to remember the slope formula: m = (change in y) / (change in x).
  • Always check your function! Plug in an x-value from the problem and see if you get the correct y-value.

🎯 Practice Suggestions

  • Create functions for real-life scenarios, like your phone data plan or how much you earn for chores.
  • Find patterns in tables and challenge yourself to write the function in under 30 seconds.
  • Use online math games that focus on linear functions and relationships.

Practice problems

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

1 2x + 5 = 15

Hint: To solve for the variable, first isolate the term containing the variable by performing the inverse operation on the constant term on the same side.

Show the answer

Answer: 5

  1. Subtract 5 from both sides of the equation. Reason: We want to isolate the term with the variable x. Since 5 is added to 2x, we do the opposite operation (subtraction) to both sides to keep the equation balanced. Calculation: 2x + 5 - 5 = 15 - 5 Result: 2x = 10
  2. Divide both sides by 2. Reason: The variable x is multiplied by 2, so we divide both sides by 2 to solve for x. Calculation: 2x / 2 = 10 / 2 Result: x = 5 Final answer: x = 5

We are solving the equation: 2x + 5 = 15

2 (3x + 15) ÷ 3 = 10

Hint: First simplify the expression by distributing the division across the terms in parentheses, then isolate the variable by performing inverse operations.

Show the answer

Answer: 5

  1. ** Understand that "÷ 3" means the same as multiplying by 1/3, so we can write: (3x + 15) / 3 = 10 **
  2. ** We can simplify the left-hand side by dividing each term in the numerator by 3: (3x)/3 + 15/3 = 10 **
  3. ** Perform the division for each term: 3x/3 = x 15/3 = 5 So we have: x + 5 = 10 **
  4. ** Isolate x by subtracting 5 from both sides: x + 5 - 5 = 10 - 5 x = 5 **
  5. ** Check the solution: Plug x = 5 into the original equation: (3*5 + 15) ÷ 3 = (15 + 15) ÷ 3 = 30 ÷ 3 = 10 This matches the right-hand side, so the solution is correct. **Final answer:** x = 5

Let's solve the equation step by step. We start with: (3x + 15) ÷ 3 = 10 **

3 (2x + 5) - (x - 3) = ?

Hint: When subtracting expressions with parentheses, distribute the negative sign to all terms inside the second set of parentheses before combining like terms.

Show the answer

Answer: x + 8

  1. Distribute the negative sign** The minus sign in front of (x - 3) means we subtract both terms inside the parentheses: (2x + 5) - x - (-3) That becomes: (2x + 5) - x + 3 **
  2. Combine like terms** First, combine the x terms: 2x - x = 1x (or just x) **
  3. Combine the constant terms** 5 + 3 = 8 **
  4. Write the final expression** x + 8 **Final Answer:** x + 8

Let's solve step by step. We start with: (2x + 5) - (x - 3) **

4 (3x + 7) - (x - 4) = ?

Hint: Remember to distribute the negative sign to all terms inside the second set of parentheses before combining like terms.

Show the answer

Answer: 2x + 11

  1. Write the original expression: (3x + 7) - (x - 4)
  2. Distribute the negative sign: 3x + 7 - x + 4
  3. Combine like terms for x: 3x - x = 2x
  4. Combine constant terms: 7 + 4 = 11
  5. Write the simplified expression: 2x + 11

The answer is 2x + 11.

5 2(3x + 5) - 4(x - 2) = 24

Hint: Distribute the coefficients to eliminate parentheses, then combine like terms to isolate the variable.

Show the answer

Answer: 3

  1. Distribute 2 into (3x + 5): 2 × 3x = 6x, 2 × 5 = 10 → 6x + 10
  2. Distribute -4 into (x - 2): -4 × x = -4x, -4 × -2 = 8 → -4x + 8
  3. Combine all terms: 6x + 10 - 4x + 8 = 24
  4. Combine like terms: (6x - 4x) = 2x, (10 + 8) = 18 → 2x + 18 = 24
  5. Subtract 18 from both sides: 2x = 6
  6. Divide both sides by 2: x = 3

The answer is 3.

6 2(3x - 5) + 4 = 3(x + 2) - 1

Hint: To solve linear equations with variables on both sides, first distribute any coefficients outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: 3

When solving equations with variables on both sides, the goal is to get all variable terms on one side and all constant terms on the other. Use inverse operations like addition, subtraction, multiplication, or division to maintain balance while simplifying. For example, in an equation like 2(a + 1) = 3(a - 2), you would distribute first, then move terms strategically.

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