Guided Practice
Work through structured problems with step-by-step guidance to reinforce function concepts and transformations.
📖 Learn
This lesson provides guided practice to help you apply what you've learned about function types and transformations. Each problem includes hints and solution steps to support your learning.
How to Use This Lesson
- Attempt each problem on your own first
- Use the hints if you get stuck
- Check your work against the provided solutions
- Note any concepts you need to review
Key Concepts Review
- Function notation: f(x) represents the output when x is the input
- Domain: All possible input values
- Range: All possible output values
- Transformations: Shifts, stretches, reflections, and compressions
💡 Worked Examples
Example 1: Identifying Function Type
Problem: Identify the type of function: f(x) = 3x^2 - 5x + 2
Solution:
- Look at the highest power of x: x^2
- The coefficient of x^2 is 3 (positive)
- This is a quadratic function that opens upward
Example 2: Describing Transformations
Problem: Describe the transformations applied to f(x) = x^2 to get g(x) = -2(x - 3)^2 + 4
Solution:
- Horizontal shift: Right 3 units (x - 3)
- Vertical stretch: Factor of 2
- Reflection: Over x-axis (negative sign)
- Vertical shift: Up 4 units (+4)
✏️ Practice Problems
Complete the following problems. Show your work for each step.
Problem 1
Identify the type of function: h(x) = 5^x
Show Hint
Look at where the variable x appears. Is it the base or the exponent?
Problem 2
Find the domain and range of f(x) = sqrt(x - 4)
Show Hint
The expression under the square root must be greater than or equal to zero.
Problem 3
Describe all transformations: g(x) = |x + 2| - 5
Show Hint
Compare to the parent function f(x) = |x|. What changed inside and outside the absolute value?
Problem 4
If f(x) = x^3, write the equation for a function that is shifted left 2 units and up 3 units.
Show Hint
Horizontal shifts affect the input (inside parentheses), vertical shifts are added outside.
Problem 5
Find f(g(x)) if f(x) = 2x + 1 and g(x) = x^2
Show Hint
Substitute g(x) for x in the function f(x).
Problem 6
Determine if the function f(x) = x^4 - x^2 is even, odd, or neither.
Show Hint
Calculate f(-x) and compare it to f(x) and -f(x).
Problem 7
Write the equation of a parabola with vertex at (2, -3) that passes through the point (4, 5).
Show Hint
Use vertex form: y = a(x - h)^2 + k, then solve for a.
Problem 8
Graph the function f(x) = -sqrt(x + 1) + 2 by identifying transformations of the parent function.
Show Hint
Start with y = sqrt(x). Apply transformations in order: horizontal shift, reflection, vertical shift.
Problem 9
Find the inverse of f(x) = (2x - 3)/5
Show Hint
Swap x and y, then solve for y.
Problem 10
The function f(x) = 2^x is transformed to g(x) = 2^(x-1) + 3. Identify the transformations and find g(2).
Show Hint
First identify the shifts, then substitute x = 2 into g(x).
✅ Check Your Understanding
Review these key questions to verify your mastery:
- Can you identify the six major function families (linear, quadratic, cubic, absolute value, square root, exponential)?
- Can you describe vertical and horizontal shifts?
- Can you identify stretches, compressions, and reflections?
- Can you find the domain and range after transformations?
- Can you compose two functions together?
🚀 Next Steps
- Review any problems where you needed hints
- Practice additional problems from your textbook
- Move on to Word Problems to see real-world applications
- Return to this page for additional practice before tests