AP SAT (Digital) Math: Quadratic & Exponential Word Problems - The Night Before 🚀

Hey! Let's get you totally prepped for those tricky quadratic and exponential word problems. No stress, just smart review. We'll break it down, connect the dots, and make sure you're feeling confident. Let's do this!

Quadratic Equations: Modeling the World 📐

Setting Up Quadratic Equations

Quadratic equations are your go-to for problems involving areas, projectile motion, and even profit. The key is to translate the word problem into the standard form: $$ax^2 + bx + c = 0$$ where $$a ≠ 0$$.

• Identify the Unknown: Assign a variable (usually x) to what you're trying to find.

• Translate: Use the given info to create your equation.

• Common Scenarios: Look for keywords like area, perimeter, revenue, or projectile motion. These often point to quadratic relationships.

Example: A garden's length is 3m more than its width, and its area is 70 sq m. If x is the width, then the equation is $$x(x+3) = 70$$.

Solving Quadratic Equations

Once you've got your equation, you've got a few ways to solve it:

• Factoring: Rewrite the quadratic as a product of two linear factors. Set each factor to zero to find solutions.
  • Example: $$x^2 + 5x + 6 = 0$$ becomes $$(x+2)(x+3) = 0$$, so x = -2 or x = -3. - Completing the Square: Transform the equation into a perfect square trinomial. Take the square root of both sides.
  • Example: $$x^2 + 6x = 16$$ becomes $$(x+3)^2 = 25$$, so x = -3 ± 5. - Quadratic Formula: The universal solver! $$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$$
  • Use it when factoring or completing the square is too messy.
  • Example: For $$2x^2 - 5x - 12 = 0$$, use a = 2, b = -5, and c = -12.
    Exam Tip

Remember the quadratic formula! It's your safety net for any quadratic equation. Write it down at the start of the exam.

Interpreting Solutions for Quadratic Word Problems

Analyzing Solution Types

• Solutions (Roots/Zeros): These are the x values that satisfy the equation.
• Discriminant: $$b^2 - 4ac$$ tells you the nature of your solutions:
  • Positive: Two distinct real solutions.
  • Zero: One repeated real solution.
  • Negative: No real solutions.

Contextual Interpretation

• Real-World Check: Solutions must make sense in the context of the problem (e.g., no negative lengths or times).
• Extraneous Solutions: Sometimes, one solution might not fit the problem's constraints. Always check!
• Example: In projectile motion, two solutions might mean the object is at a certain height twice (once going up, once going down).

Exponential Functions: Growth and Decay 📈

Identifying Exponential Models

Exponential functions model situations where there's a constant rate of growth or decay. The general form is $$f(x) = a * b^x$$

• a (Initial Value): The starting amount when x = 0. - b (Growth/Decay Factor):

  • b > 1: Growth (e.g., 1.05 = 5% growth).

  • 0 < b < 1: Decay (e.g., 0.9 = 10% decay).

Example: A population starts at 1000 and grows 2% annually: $$P(t) = 1000 * 1.02^t$$, where t is in years.

Applications of Exponential Models

Exponential functions pop up everywhere:

• Population Growth: $$N(t) = N_0 * 2^t$$ (doubling every time period).
• Compound Interest: $$A(t) = P * (1 + r)^t$$ (P = principal, r = rate, t = time).
• Radioactive Decay: $$A(t) = A_0 * (0.5)^{t/h}$$ (h = half-life).
• Disease Spread: $$I(t) = I_0 * (1 + r)^t$$ (r = infection rate).

Solving Exponential Word Problems

Applying Exponent Properties

• Product Rule: $$a^m * a^n = a^{m+n}$$

• Quotient Rule: $$a^m / a^n = a^{m-n}$$

• Power Rule: $$(a^m)^n = a^{mn}$$

• Zero Exponent: $$a^0 = 1$$

• Negative Exponent: $$a^{-n} = 1 / a^n$$

Utilizing Logarithms

• Logarithms are the inverse of exponentials: If $$a^x = b$$, then $$log_a(b) = x$$.
• Log Rules:
  • Product: $$log_a(M * N) = \log_a(M) + \log_a(N)$$
  • Quotient: $$log_a(M / N) = \log_a(M) - \log_a(N)$$
  • Power: $$log_a(M^n) = n * \log_a(M)$$
  • Change of Base: $$log_a(x) = \frac{\log_b(x)}{\log_b(a)}$$

Example: Solve $$2^x = 32$$ - Take $$\log_2$$ of both sides: $$\log_2(2^x) = \log_2(32)$$ - Power rule: $$x * \log_2(2) = \log_2(32)$$ - Simplify: $$x = 5$$

Final Exam Focus 🎯

• High-Priority Topics:
  • Setting up and solving quadratic equations (especially with word problems).
  • Understanding exponential growth and decay models.
  • Using logarithms to solve exponential equations.
• Common Question Types:
  • Area and perimeter problems involving quadratics.
  • Projectile motion problems (height as a function of time).
  • Population growth and compound interest problems.
  • Radioactive decay and half-life problems.
• Time Management:
  • Quickly identify the type of problem (quadratic or exponential).
  • Set up the equation accurately.
  • Solve efficiently using the appropriate method.
• Common Pitfalls:
  • Forgetting to check if solutions make sense in context.
  • Mixing up growth and decay factors.
  • Incorrectly applying logarithm rules.

Practice Question

Practice Questions

Multiple Choice Questions

1. A ball is thrown vertically upward from a height of 5 feet with an initial velocity of 40 feet per second. The height of the ball, h, after t seconds is given by the equation $$h = -16t^2 + 40t + 5$$. At what time will the ball hit the ground? (A) 0.12 seconds (B) 2.62 seconds (C) 1.25 seconds (D) 5.25 seconds

2. A population of bacteria doubles every 3 hours. If there are initially 200 bacteria, how many bacteria will there be after 12 hours? (A) 800 (B) 1600 (C) 3200 (D) 6400

Free Response Question

A rectangular garden is to be enclosed by 100 feet of fencing. Let x represent the width of the garden.

(a) Express the length of the garden in terms of x. (b) Write an expression for the area of the garden in terms of x. (c) Find the value of x that maximizes the area of the garden. (d) What is the maximum area of the garden?

Scoring Breakdown:

(a) (2 points) - 1 point for recognizing that the perimeter is 2length + 2width = 100 - 1 point for expressing length as 50 - x (b) (2 points) - 1 point for using the formula Area = length * width - 1 point for writing the area as A = x(50-x) or A = 50x - x^2 (c) (3 points) - 1 point for recognizing that the maximum occurs at the vertex of the parabola - 1 point for using -b/2a to find the x-coordinate of the vertex - 1 point for correctly calculating x = 25 (d) (2 points) - 1 point for substituting x = 25 into the area equation - 1 point for calculating the maximum area as 625 square feet

Remember, you've got this! Take a deep breath, trust your preparation, and go ace that exam! 🌟