College Algebra Test 2 Review: Quadratic Functions, Vertex, Discriminant, and Applications

Q1. For the function $f(x) = 2x^2 + 7x + 3$:

Background

Topic: Quadratic Functions

This question tests your understanding of quadratic functions, including their standard form, axis of symmetry, vertex, discriminant, and how to interpret these features graphically and algebraically.

Key Terms and Formulas:

• Standard form of a quadratic: $f(x) = ax^2 + bx + c$

• Axis of symmetry: $x = -\frac{b}{2a}$

• Vertex: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

• Discriminant: $D = b^2 - 4ac$

Step-by-Step Guidance

1. Identify the coefficients: $a = 2$, $b = 7$, $c = 3$ for the quadratic function.

2. Write the equation for the axis of symmetry using $x = -\frac{b}{2a}$ and substitute the values for $b$ and $a$.

3. To find the vertex, use the axis of symmetry value for $x$ and substitute it back into $f(x)$ to find the $y$-coordinate.

4. Calculate the discriminant $D = b^2 - 4ac$ to determine the nature of the roots of the function.

5. Interpret the discriminant: If $D > 0$, there are two real roots; if $D = 0$, one real root; if $D < 0$, no real roots.

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Q2. A factory purchases a machine for $150,000. It is depreciated using the straight-line method over 10 years.

Background

Topic: Linear Functions and Applications

This question tests your ability to model depreciation using linear functions, interpret slope and y-intercept, and analyze function behavior over time.

Key Terms and Formulas:

• Straight-line depreciation formula: $V(t) = V_0 - rt$

• $V_0$ = initial value, $r$ = rate of depreciation per year, $t$ = time in years

Step-by-Step Guidance

1. Identify the initial value ($V_0 = 150,000$) and the time period (10 years).

2. Calculate the annual depreciation rate: $r = \frac{V_0}{10}$.

3. Write the linear function for the machine's value over time: $V(t) = 150,000 - rt$.

4. To find the book value after 7 years, substitute $t = 7$ into your function.

5. Determine the domain and range of the function based on the context (years and value).

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Q3. An electronics company has found that when certain calculators are sold at a price of $p$ dollars per unit, the number of calculators sold is given by the demand equation $q = 750 - 6p$.

Background

Topic: Linear Demand Functions

This question tests your ability to interpret and manipulate linear demand equations, analyze slope and intercept, and understand maximum revenue concepts.

Key Terms and Formulas:

• Demand function: $q = 750 - 6p$

• Revenue function: $R = p \cdot q$

Step-by-Step Guidance

1. Identify the demand function and understand what $q$ and $p$ represent.

2. Write the revenue function in terms of $p$: $R(p) = p \cdot (750 - 6p)$.

3. Expand the revenue function to standard quadratic form: $R(p) = 750p - 6p^2$.

4. To find the price that maximizes revenue, use the vertex formula for a quadratic: $p = -\frac{b}{2a}$, where $a$ and $b$ are coefficients from $R(p)$.

5. Substitute this value of $p$ back into $R(p)$ to find the maximum revenue.

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