Circle and Parabola Test Practice Problems – Step-by-Step Precalculus Guidance

Q1. Find the length and midpoint of the segment connecting (-3, 7) and (-9, 4).

Background

Topic: Distance and Midpoint Formulas

This question tests your understanding of how to calculate the distance between two points and the midpoint of a segment in the coordinate plane.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Label the points: $A(-3, 7)$ and $B(-9, 4)$.

2. For the distance, substitute the coordinates into the distance formula: $d = \sqrt{(-9 - (-3))^2 + (4 - 7)^2}$.

3. Simplify inside the parentheses: $(-9 - (-3)) = -6$ and $(4 - 7) = -3$.

4. For the midpoint, substitute into the midpoint formula: $M = \left( \frac{-3 + (-9)}{2}, \frac{7 + 4}{2} \right)$.

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Q2. Write the equation of a circle with center (0, -5) and radius 7.

Background

Topic: Equation of a Circle

This question tests your ability to write the equation of a circle given its center and radius.

Key Terms and Formulas:

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

• Where $(h, k)$ is the center and $r$ is the radius.

Step-by-Step Guidance

1. Identify the center $(h, k) = (0, -5)$ and radius $r = 7$.

2. Substitute these values into the standard form: $(x - 0)^2 + (y - (-5))^2 = 7^2$.

3. Simplify the equation as much as possible.

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Q3. Write the equation of a circle with endpoints of a diameter at (-9, 2) and (1, 4).

Background

Topic: Equation of a Circle from Diameter Endpoints

This question tests your ability to find the center and radius of a circle when given the endpoints of its diameter.

Key Terms and Formulas:

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

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

• Radius is half the diameter.

• Circle equation: $(x - h)^2 + (y - k)^2 = r^2$

Step-by-Step Guidance

1. Find the center by calculating the midpoint of $(-9, 2)$ and $(1, 4)$.

2. Find the length of the diameter using the distance formula.

3. Divide the diameter by 2 to get the radius.

4. Substitute the center and radius into the standard form of the circle equation.

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Q4. Determine the center and radius of the circle $x^2 + y^2 - 10x + 4y - 9 = 0$.

Background

Topic: Completing the Square for Circles

This question tests your ability to rewrite a general equation of a circle in standard form by completing the square.

Key Terms and Formulas:

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

• Complete the square for $x$ and $y$ terms.

Step-by-Step Guidance

1. Group $x$ and $y$ terms: $(x^2 - 10x) + (y^2 + 4y) = 9$.

2. Complete the square for $x$ and $y$ terms separately.

3. Add the necessary constants to both sides to balance the equation.

4. Rewrite in standard form to identify the center $(h, k)$ and radius $r$.

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Q5. Which points are on the circle $x - 12 + y + 32 = 100$?

Background

Topic: Points on a Circle

This question tests your ability to determine if a point lies on a given circle by substituting the coordinates into the equation.

Key Terms and Formulas:

• Substitute $(x, y)$ into the equation and check if the equation is satisfied.

Step-by-Step Guidance

1. For each point, substitute $x$ and $y$ into the equation $x - 12 + y + 32 = 100$.

2. Simplify to see if the left side equals 100.

3. Repeat for all points.

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Q6. Write the equation of a parabola with a maximum value of -7 at $x = 5$ and passing through (3, -19). Leave your answer in vertex form.

Background

Topic: Vertex Form of a Parabola

This question tests your ability to write the equation of a parabola given its vertex and a point it passes through.

Key Terms and Formulas:

• Vertex form: $y = a(x - h)^2 + k$

• Vertex: $(h, k)$

Step-by-Step Guidance

1. Identify the vertex: $(h, k) = (5, -7)$.

2. Substitute the vertex into the vertex form: $y = a(x - 5)^2 - 7$.

3. Substitute the point $(3, -19)$ into the equation to solve for $a$.

4. Solve for $a$ but do not substitute back yet.

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Q7. An object is thrown vertically upward from the edge of a 240-foot building. The function models the height of the object in feet $t$ seconds after it was thrown. What is the maximum height of the object?

Background

Topic: Quadratic Functions – Maximum Value

This question tests your ability to find the maximum value of a quadratic function, which represents the object's maximum height.

Key Terms and Formulas:

• Quadratic function: $h(t) = at^2 + bt + c$

• Maximum occurs at $t = -\frac{b}{2a}$ for $a < 0$.

• Substitute $t$ back into $h(t)$ to find the maximum height.

Step-by-Step Guidance

1. Identify the coefficients $a$, $b$, and $c$ from the given function.

2. Calculate $t = -\frac{b}{2a}$ to find the time at which the maximum occurs.

3. Substitute this $t$ value back into the function to find the maximum height.

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Q8. Quadratic Functions: For each function, determine the vertex, axis of symmetry, max/min value, domain, range, y-intercept, intervals of increase, and x-intercepts.

Background

Topic: Analyzing Quadratic Functions

This question tests your ability to analyze the properties of quadratic functions, including their graphs and key features.

Key Terms and Formulas:

• Vertex: $x = -\frac{b}{2a}$, $y = f(x)$

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

• Domain: All real numbers

• Range: Depends on whether the parabola opens up or down

• y-intercept: Set $x = 0$

• x-intercepts: Set $y = 0$ and solve for $x$

Step-by-Step Guidance

1. For each function, identify $a$, $b$, and $c$.

2. Calculate the vertex using $x = -\frac{b}{2a}$ and $y = f(x)$.

3. Write the equation of the axis of symmetry.

4. Determine if the vertex is a maximum or minimum based on the sign of $a$.

5. Find the y-intercept by evaluating $f(0)$.

6. Set $f(x) = 0$ and solve for $x$ to find the x-intercepts.

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Q9. You have a 500-foot roll of fencing and a large field. You want to fence a rectangular playground. What are the dimensions of the largest playground you can fence? What is the largest area?

Background

Topic: Optimization with Quadratic Functions

This question tests your ability to use quadratic functions to maximize area given a fixed perimeter.

Key Terms and Formulas:

• Let $L$ = length and $w$ = width.

• Perimeter: $2L + 2w = 500$

• Area: $A = L \times w$

Step-by-Step Guidance

1. Express one variable in terms of the other using the perimeter equation: $2L + 2w = 500$.

2. Solve for $L$ in terms of $w$ (or vice versa).

3. Substitute this expression into the area formula to get $A$ as a function of one variable.

4. Write $A(w)$ as a quadratic function and identify the value of $w$ that maximizes $A$.

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Q10. Factor the following expressions:

Background

Topic: Factoring Quadratic and Higher Degree Polynomials

This question tests your ability to factor various types of polynomials, including quadratics and higher degree expressions.

Key Terms and Formulas:

• Factoring quadratics: $ax^2 + bx + c = (x + m)(x + n)$ where $m$ and $n$ are numbers that multiply to $ac$ and add to $b$.

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

• Factoring by grouping for higher degree polynomials.

Step-by-Step Guidance

1. For each expression, look for common factors first.

2. For quadratics, identify $a$, $b$, and $c$ and find two numbers that multiply to $ac$ and add to $b$.

3. For special forms (like $x^2 - 9$), recognize patterns such as difference of squares.

4. For higher degree polynomials, try factoring by grouping or other methods as appropriate.

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Q11. Graph the circle $x^2 + 8x + y^2 - 14y + 61 = 0$.

Background

Topic: Graphing Circles by Completing the Square

This question tests your ability to rewrite the equation of a circle in standard form and graph it on the coordinate plane.

Key Terms and Formulas:

• Complete the square for $x$ and $y$ terms to rewrite in standard form: $(x - h)^2 + (y - k)^2 = r^2$

• Identify the center $(h, k)$ and radius $r$.

Step-by-Step Guidance

1. Group $x$ and $y$ terms: $(x^2 + 8x) + (y^2 - 14y) = -61$.

2. Complete the square for both $x$ and $y$ terms.

3. Add the necessary constants to both sides to balance the equation.

4. Identify the center and radius from the standard form.

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Q12. Write the equation of the circle whose diameter has endpoints (-5, 3) and (-7, 33).

Background

Topic: Equation of a Circle from Diameter Endpoints

This question tests your ability to find the center and radius of a circle given the endpoints of its diameter.

Key Terms and Formulas:

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

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

• Radius is half the diameter.

• Circle equation: $(x - h)^2 + (y - k)^2 = r^2$

Step-by-Step Guidance

1. Find the center by calculating the midpoint of $(-5, 3)$ and $(-7, 33)$.

2. Find the length of the diameter using the distance formula.

3. Divide the diameter by 2 to get the radius.

4. Substitute the center and radius into the standard form of the circle equation.

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Q13. Write $y = x^2 - 8x + 10$ in vertex form. Name the vertex and y-intercept.

Background

Topic: Completing the Square for Parabolas

This question tests your ability to rewrite a quadratic in vertex form and identify key features.

Key Terms and Formulas:

• Vertex form: $y = a(x - h)^2 + k$

• Complete the square to convert from standard to vertex form.

• y-intercept: Set $x = 0$.

Step-by-Step Guidance

1. Group $x$ terms: $y = (x^2 - 8x) + 10$.

2. Complete the square for $x^2 - 8x$.

3. Rewrite the equation in vertex form.

4. Identify the vertex $(h, k)$ and y-intercept.

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Q14. Write $y = 12x + 62 + 1$ in standard form. Name the vertex and y-intercept.

Background

Topic: Standard Form of a Quadratic

This question tests your ability to rewrite a quadratic in standard form and identify key features.

Key Terms and Formulas:

• Standard form: $y = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• y-intercept: Set $x = 0$.

Step-by-Step Guidance

1. Combine like terms to write the equation in standard form.

2. Identify $a$, $b$, and $c$.

3. Find the vertex using $x = -\frac{b}{2a}$ and $y = f(x)$.

4. Find the y-intercept by evaluating $f(0)$.

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Q15. The parabola is written in factored form: $y = 3x + 1x - 5$

Background

Topic: Factored and Standard Forms of Quadratics

This question tests your ability to interpret and rewrite quadratic equations from factored to standard form.

Key Terms and Formulas:

• Factored form: $y = a(x - r_1)(x - r_2)$

• Standard form: $y = ax^2 + bx + c$

• Roots (x-intercepts) are $r_1$ and $r_2$.

Step-by-Step Guidance

1. Identify the roots from the factored form.

2. Expand the expression to write it in standard form.

3. Identify the coefficients $a$, $b$, and $c$.

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