Worked Examples: Advanced Math

Example 1: Quadratic Modeling (Medium)

A ball is launched from ground level. Its height $h$ in feet after $t$ seconds is modeled by $h = -16t^2 + 64t$. After how many seconds does the ball return to the ground?

Solution: Set $h = 0$: $-16t^2 + 64t = 0$

Factor out $-16t$: $-16t(t - 4) = 0$

$t = 0$ (launch) or $t = 4$ (landing).

Answer: 4 seconds.

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Example 2: Vertex and Maximum (Medium–Hard)

Using the same model $h = -16t^2 + 64t$, what is the maximum height the ball reaches?

Solution: Method 1 (Completing the square): $h = -16(t^2 - 4t) = -16(t^2 - 4t + 4 - 4) = -16(t - 2)^2 + 64$

Maximum height is $64$ feet, reached at $t = 2$ seconds.

Method 2 (Vertex formula): $t = -b/(2a) = -64/(2 \cdot -16) = 2$. Then $h(2) = -16(4) + 64(2) = 64$.

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Example 3: Exponential Context (Medium)

An investment of $5{,}000 earns 4$t$ years?

Solution: Initial value: $5{,}000$. Growth factor: $1 + 0.04 = 1.04$.

$V = 5{,}000 \cdot (1.04)^t$

SAT follow-up: "What is the value after 3 years?"

$V = 5{,}000 \cdot (1.04)^3 = 5{,}000 \cdot 1.124864 = 5{,}624.32$

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Example 4: Discriminant Question (Hard)

For what value of $c$ does $2x^2 - 8x + c = 0$ have exactly one real solution?

Solution: Set discriminant to zero: $b^2 - 4ac = 0$

$(-8)^2 - 4(2)(c) = 0$

$64 - 8c = 0$

$c = 8$

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Example 5: Nonlinear System (Hard)

How many solutions does the system $y = 2x + 3$ and $y = x^2 + 2x - 1$ have?

Solution: Set equal: $2x + 3 = x^2 + 2x - 1$

Simplify: $0 = x^2 - 4$

Factor: $(x-2)(x+2) = 0$, so $x = 2$ or $x = -2$.

Check $y$-values: $y = 7$ and $y = -1$. Two distinct intersection points.

Answer: 2 solutions.