Step-by-Step Precalculus Midterm Study Guidance

Q1. A backyard is composed of a garden of size 4 ft x 10 ft, surrounded by a path of uniform width. If the total area of the yard is 112 ft², what is the width of the walkway? (Algebraic solution required.)

Background

Topic: Quadratic Equations & Geometry

This question tests your ability to set up and solve a quadratic equation based on geometric relationships involving area.

Key Terms and Formulas:

• Area of rectangle: $A = l \times w$

• Let $x$ be the width of the walkway.

• Garden size: $4 \text{ ft} \times 10 \text{ ft}$

• Total area (including walkway): $112 \text{ ft}^2$

Step-by-Step Guidance

1. Let $x$ be the width of the walkway. The overall length and width of the yard will be increased by $2x$ (since the walkway surrounds the garden on all sides).

2. Write expressions for the total length and width: $\text{Total length} = 10 + 2x$, $\text{Total width} = 4 + 2x$.

3. Set up the equation for the total area: $(10 + 2x)(4 + 2x) = 112$.

4. Expand the left side and simplify: $40 + 20x + 8x + 4x^2 = 112$.

5. Combine like terms and rearrange to form a quadratic equation.

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Q2. Solve and check your answers: $\frac{x}{x-7} - \frac{7}{x+7} = \frac{14x}{x^2-49}$

Background

Topic: Rational Equations

This question tests your ability to solve rational equations and check for extraneous solutions.

Key Terms and Formulas:

• Rational equation: An equation involving fractions with polynomials in the numerator and denominator.

• Common denominator: $x^2 - 49 = (x-7)(x+7)$

Step-by-Step Guidance

1. Identify the least common denominator (LCD): $x^2 - 49$.

2. Multiply both sides of the equation by the LCD to clear denominators.

3. Expand and simplify each term after multiplying.

4. Combine like terms and rearrange to form a linear or quadratic equation.

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Q3. Solve and check your answers: $\sqrt{6x-11} - 2x = -7$

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots and check for extraneous solutions.

Key Terms and Formulas:

• Radical equation: An equation with a variable inside a square root.

• To isolate the radical, add $2x$ to both sides.

Step-by-Step Guidance

1. Isolate the square root: $\sqrt{6x-11} = 2x - 7$.

2. Square both sides to eliminate the radical: $6x - 11 = (2x - 7)^2$.

3. Expand the right side: $(2x - 7)^2 = 4x^2 - 28x + 49$.

4. Move all terms to one side to form a quadratic equation.

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Q4. Solve using substitution: $(x^2 - 1)^2 - 8(x^2 - 1) - 20 = 0$

Background

Topic: Quadratic Equations & Substitution

This question tests your ability to use substitution to simplify and solve a quartic equation.

Key Terms and Formulas:

• Let $u = x^2 - 1$

• Quadratic form: $u^2 - 8u - 20 = 0$

Step-by-Step Guidance

1. Let $u = x^2 - 1$ and rewrite the equation in terms of $u$.

2. Now solve the quadratic equation: $u^2 - 8u - 20 = 0$.

3. Factor or use the quadratic formula to solve for $u$.

4. Substitute back $x^2 - 1$ for $u$ and solve for $x$.

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Q5. Solve the following rational inequality using the test-point method, including an appropriate sign chart. Write your answer using interval notation: $\frac{x^2 - 2x - 35}{x^2 - 16} \geq 0$

Background

Topic: Rational Inequalities & Test-Point Method

This question tests your ability to solve rational inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Numerator: $x^2 - 2x - 35$

• Denominator: $x^2 - 16$

• Critical points: Where numerator or denominator equals zero.

Step-by-Step Guidance

1. Factor the numerator and denominator: $x^2 - 2x - 35 = (x - 7)(x + 5)$, $x^2 - 16 = (x - 4)(x + 4)$.

2. Identify the critical points: $x = 7, -5, 4, -4$.

3. Draw a number line and mark the critical points.

4. Use the test-point method to determine the sign of the expression in each interval.

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Q6. Solve the inequality. Illustrate your solution on the real number line and express in interval notation: $\frac{1}{4}|2x+1| + 1 \geq 13$

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute value and represent the solution graphically and in interval notation.

Key Terms and Formulas:

• Absolute value: $|2x+1|$

• To isolate the absolute value, subtract 1 and multiply both sides by 4.

Step-by-Step Guidance

1. Subtract 1 from both sides: $\frac{1}{4}|2x+1| \geq 12$

2. Multiply both sides by 4: $|2x+1| \geq 48$

3. Set up two inequalities: $2x+1 \geq 48$ and $2x+1 \leq -48$

4. Solve each inequality for $x$.

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Q7. Given the points $P(-5,0)$ and $Q(1,-3)$, find the midpoint of the line segment from $P$ to $Q$ and the distance from point $P$ to point $Q$.

Background

Topic: Coordinate Geometry

This question tests your ability to use midpoint and distance formulas in the coordinate plane.

Key Terms and Formulas:

• Midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Guidance

1. Plug the coordinates into the midpoint formula: $M = \left( \frac{-5 + 1}{2}, \frac{0 + (-3)}{2} \right)$

2. Plug the coordinates into the distance formula: $d = \sqrt{(1 - (-5))^2 + (-3 - 0)^2}$

3. Simplify the expressions for both midpoint and distance.

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Q8. For the circle $x^2 + y^2 + 6x + 10y = 4$:

Background

Topic: Circles & Completing the Square

This question tests your ability to rewrite the equation of a circle in standard form and find intercepts.

Key Terms and Formulas:

• Standard form: $(x-h)^2 + (y-k)^2 = r^2$

• Completing the square for $x$ and $y$ terms

Step-by-Step Guidance

1. Group $x$ and $y$ terms: $x^2 + 6x$ and $y^2 + 10y$

2. Complete the square for each group.

3. Rewrite the equation in standard form and identify $h$, $k$, and $r$.

4. To find intercepts, set $x=0$ and $y=0$ and solve for the other variable.

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Q9. Find the domain of the function $g(x) = \frac{\sqrt{x+11}}{3x-1}$ and express using interval notation.

Background

Topic: Domain of Functions

This question tests your ability to find the domain of a function involving a square root and a rational denominator.

Key Terms and Formulas:

• Domain: Set of all $x$ values for which the function is defined.

• Square root: $x+11 \geq 0$

• Denominator: $3x-1 \neq 0$

Step-by-Step Guidance

1. Set $x+11 \geq 0$ to ensure the square root is defined.

2. Set $3x-1 \neq 0$ to avoid division by zero.

3. Combine the two conditions to find the domain.

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Q10. Find the average rate of change of the function $f(x) = \frac{5}{4x-1}$ from $x = -1$ to $x = 1$.

Background

Topic: Average Rate of Change

This question tests your ability to compute the average rate of change of a function over a given interval.

Key Terms and Formulas:

• Average rate of change: $\frac{f(b) - f(a)}{b - a}$

• Here, $a = -1$, $b = 1$

Step-by-Step Guidance

1. Calculate $f(-1)$ and $f(1)$ by plugging into the function.

2. Set up the average rate of change formula: $\frac{f(1) - f(-1)}{1 - (-1)}$

3. Simplify the numerator and denominator.

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Q11. Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x) = -4x^2 + 7x - 1$.

Background

Topic: Difference Quotient & Algebraic Manipulation

This question tests your ability to simplify the difference quotient for a quadratic function.

Key Terms and Formulas:

• Difference quotient: $\frac{f(x+h)-f(x)}{h}$

• Plug $x+h$ into the function and expand.

Step-by-Step Guidance

1. Compute $f(x+h)$: $-4(x+h)^2 + 7(x+h) - 1$

2. Expand $f(x+h)$ and subtract $f(x)$.

3. Simplify the numerator and divide by $h$.

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Q12. For the functions $f(x) = \frac{3x+4}{x-2}$ and $g(x) = \frac{1}{x-2}$:

Background

Topic: Function Operations & Composition

This question tests your ability to evaluate function products and compositions, and find domains.

Key Terms and Formulas:

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Composition: $(f \circ g)(x) = f(g(x))$

• Domain: Values of $x$ for which the function is defined.

Step-by-Step Guidance

1. For $(fg)(6)$, evaluate $f(6)$ and $g(6)$, then multiply.

2. For $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$ wherever $x$ appears.

3. Find the domain by considering restrictions from both $f$ and $g$.

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Q13. Sketch a graph of the function. Find the standard form $f(x) = a(x-h)^2 + k$ of the function, identify the vertex and include all intercepts if they exist. Show algebraic work. Identify all points as ordered pairs beside the appropriate point on the graph. $g(x) = -2x^2 -16x + 17$

Background

Topic: Quadratic Functions & Graphing

This question tests your ability to rewrite a quadratic in standard form, identify the vertex, and find intercepts.

Key Terms and Formulas:

• Standard form: $f(x) = a(x-h)^2 + k$

• Vertex: $(h, k)$

• Intercepts: Set $x=0$ for $y$-intercept, $g(x)=0$ for $x$-intercepts

Step-by-Step Guidance

1. Complete the square to rewrite $g(x)$ in standard form.

2. Identify the vertex $(h, k)$ from the standard form.

3. Find the $y$-intercept by plugging $x=0$ into $g(x)$.

4. Find the $x$-intercepts by solving $g(x)=0$.

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Q14. For the one-to-one function $f(x) = \frac{6x-5}{4x+11}$:

Background

Topic: Inverse Functions & Domain/Range

This question tests your ability to find the inverse of a rational function and determine domain and range.

Key Terms and Formulas:

• Inverse function: Swap $x$ and $y$ and solve for $y$.

• Domain: Values of $x$ for which $f(x)$ is defined.

• Range: Values of $y$ that $f(x)$ can take.

Step-by-Step Guidance

1. Let $y = \frac{6x-5}{4x+11}$, then swap $x$ and $y$ to get $x = \frac{6y-5}{4y+11}$.

2. Solve for $y$ in terms of $x$ to find $f^{-1}(x)$.

3. Find domain and range by considering restrictions on $x$ and $y$.

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Q15. Sketch the polynomial function $P(x) = x^2(x^2 + 1)(x^2 - 36)(x+2)^4$. Identify the end behavior, include all intercepts as ordered pairs on the graph, and indicate the multiplicity at each zero.

Background

Topic: Polynomial Functions & Graphing

This question tests your ability to analyze and sketch a polynomial, including degree, end behavior, intercepts, and multiplicity.

Key Terms and Formulas:

• Degree: Highest power of $x$ in the expanded form.

• End behavior: Determined by degree and leading coefficient.

• Multiplicity: Number of times a zero is repeated.

• Intercepts: Set $x=0$ for $y$-intercept, $P(x)=0$ for $x$-intercepts

Step-by-Step Guidance

1. Expand the factors to determine the degree and leading coefficient.

2. Identify all real zeros and their multiplicities.

3. Find the $y$-intercept by plugging $x=0$ into $P(x)$.

4. Describe the end behavior based on degree and leading coefficient.

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