Math Practice Test

Key Concepts and Operations

Operations with Functions

• G(-2) - H(-2):
  • G(-2) = 2(-2) - 2 = -4 - 2 = -6.
  • H(-2) = (-2)^2 + (-2) = 4 - 2 = 2.
  • So, G(-2) - H(-2) = -6 - 2 = -8.

Difference of Functions

• H(t) - G(t):
  • Given H(t) = -t + 1 and G(t) = 4t + 2.
  • Combine like terms:
    • (-t - 4t) = -5t.
    • (1 - 2) = -1.
  • Result: -5t - 1.

Substitution in Functions

• F - G with n/2:
  • Substitute n/2 into the functions:
    • F(n/2) = 4(n/2) - 3.
    • G(n/2) = 3 + 5(n/2)^2.
  • Simplify the expressions and find like terms.

Graphing Functions Domains

• Piecewise Function:
  • For x <= 0: y = -x - 1 (slope: -1).
  • For 0 < x < 4: y = -4 (y-constant across the domain).
  • For x >= 4: y = -2.

Transformations of Graphs

• Graph Translations and Dilations:
  • Translation: Shift the graph down by 2 units.
  • Dilation or compression: Scale the graph vertically by a factor (e.g., 1/3).

Square Root and Absolute Value Transformations

• Graph Reflections:
  • Reflection across the y-axis or x-axis depending on inside/outside negative signs.
• Absolute Value Function:
  • New zero at x = 3.

Quadratic Functions and Graphs

• Vertex and Intercepts: Quadratic Equations:
  • Determine vertices from symmetrical points (midpoints between intercepts).
  • Use standard form and factorization to find x-intercepts.

Analyzing Quadratic Inequalities

• Inequality Solving:
  • Determine where a quadratic is less than or equal to zero by testing points in different segments of the domain (using test points around critical values).
  • Utilize endpoints of test intervals based on where the function equals zero.

Practice with Graphs and Equations

• Sketch and Identify:
  • Sketch quadratics based on intercepts and vertex.
  • Identify x-intercepts as solutions of the quadratic equation.

Considerations for Function Behavior

• Increasing and Decreasing Intervals:
  • Identify where functions are increasing or decreasing by analyzing the graph over intervals.
  • Consider continuity and smoothness of functions in transition.