Precalculus Study Guide: Conics (Parabolas, Ellipses, Hyperbolas)

Q1. Graph the following parabola. Make sure to answer the following and display them on the graph. Make sure the equation is in standard form.

Background

Topic: Parabolas (Conic Sections)

This question is testing your understanding of the properties of parabolas, including how to find the vertex, focus, directrix, and how to write the equation in standard form. You are also asked to graph the parabola and label key features.

Key Terms and Formulas

• Standard form for a vertical parabola: $(x-h)^2 = 4p(y-k)$

• Vertex: $(h, k)$

• Focus: $(h, k+p)$

• Directrix: $y = k-p$

• p: Distance from vertex to focus (and from vertex to directrix)

Step-by-Step Guidance

1. Identify the vertex of the parabola from the given information. For example, if the vertex is $(1, -5)$, set $h = 1$ and $k = -5$.

2. Determine the value of $p$ using the given or calculated information. If $4p = -8$, solve for $p$ by dividing both sides by 4.

3. Write the equation of the parabola in standard form using the values of $h$, $k$, and $p$.

4. Find the coordinates of the focus and the equation of the directrix using the formulas above. Remember, the focus is $p$ units from the vertex in the direction the parabola opens, and the directrix is $p$ units in the opposite direction.

5. Set up the equation to find the intersection with the y-axis (if needed) by substituting $x = 0$ into your standard form equation. Solve for $y$ but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer:

Vertex: $(1, -5)$ Standard form: $(x-1)^2 = -8(y+5)$ Focus: $(1, -7)$ Directrix: $y = -3$ Y-intercept: $(0, -13)$ The parabola opens downward because $p$ is negative.

All key features are labeled and the equation is in standard form as required.

Q2. Graph the following parabola. Make sure to answer the following and display them on the graph. Make sure the equation is in standard form.

Background

Topic: Parabolas (Conic Sections)

This question is similar to the previous one but involves a parabola that opens to the right. You are asked to find the vertex, focus, directrix, and write the equation in standard form.

Key Terms and Formulas

• Standard form for a horizontal parabola: $(y-k)^2 = 4p(x-h)$

• Vertex: $(h, k)$

• Focus: $(h+p, k)$

• Directrix: $x = h-p$

• p: Distance from vertex to focus (and from vertex to directrix)

Step-by-Step Guidance

1. Identify the vertex of the parabola from the given information. For example, if the vertex is $(3, -4)$, set $h = 3$ and $k = -4$.

2. Determine the value of $p$ using the given or calculated information. If $4p = 16$, solve for $p$ by dividing both sides by 4.

3. Write the equation of the parabola in standard form using the values of $h$, $k$, and $p$.

4. Find the coordinates of the focus and the equation of the directrix using the formulas above. The focus is $p$ units from the vertex in the direction the parabola opens, and the directrix is $p$ units in the opposite direction.

5. Set up the equation to find the intersection with the x-axis (if needed) by substituting $y = 0$ into your standard form equation. Solve for $x$ but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer:

Vertex: $(3, -4)$ Standard form: $(y+4)^2 = 16(x-3)$ Focus: $(7, -4)$ Directrix: $x = -1$ X-intercept: $(7, 0)$ The parabola opens to the right because $p$ is positive.

Q3. Graph the ellipse. Make sure to answer the following and clearly display them on the graph. Make sure the equation is in standard form.

Background

Topic: Ellipses (Conic Sections)

This question is testing your ability to write the equation of an ellipse in standard form, identify the center, vertices, co-vertices, and foci, and graph the ellipse with all key features labeled.

Key Terms and Formulas

• Standard form for an ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (if $a > b$, major axis is horizontal)

• Center: $(h, k)$

• Vertices: $(h \pm a, k)$ or $(h, k \pm b)$

• Co-vertices: $(h, k \pm b)$ or $(h \pm a, k)$

• Foci: $(h \pm c, k)$ or $(h, k \pm c)$, where $c = \sqrt{a^2 - b^2}$

Step-by-Step Guidance

1. Write the equation of the ellipse in standard form. Identify $h$, $k$, $a^2$, and $b^2$ from the equation.

2. Find the center of the ellipse, which is $(h, k)$.

3. Calculate the lengths of the major and minor axes using $a$ and $b$.

4. Find the coordinates of the vertices and co-vertices using the values of $a$ and $b$.

5. Calculate $c = \sqrt{a^2 - b^2}$ to find the foci, but stop before plugging in the final values.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x-2)^2}{7} + \frac{(y-2)^2}{12} = 1$ Center: $(2, 2)$ Vertices: $(2, 2 \pm \sqrt{12})$ Co-vertices: $(2 \pm \sqrt{7}, 2)$ Foci: $(2, 2 \pm \sqrt{5})$ All features are labeled and the equation is in standard form.

Q4. Graph the hyperbola. Make sure to answer the following and display them on the graph. Make sure the equation is in standard form.

Background

Topic: Hyperbolas (Conic Sections)

This question is testing your ability to write the equation of a hyperbola in standard form, identify the center, vertices, foci, and asymptotes, and graph the hyperbola with all key features labeled.

Key Terms and Formulas

• Standard form for a hyperbola (vertical transverse axis): $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

• Standard form for a hyperbola (horizontal transverse axis): $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Center: $(h, k)$

• Vertices: $(h, k \pm a)$ or $(h \pm a, k)$

• Foci: $(h, k \pm c)$ or $(h \pm c, k)$, where $c = \sqrt{a^2 + b^2}$

• Asymptotes: $y - k = \pm \frac{a}{b}(x - h)$ or $y - k = \pm \frac{b}{a}(x - h)$

Step-by-Step Guidance

1. Write the equation of the hyperbola in standard form. Identify $h$, $k$, $a^2$, and $b^2$ from the equation.

2. Find the center of the hyperbola, which is $(h, k)$.

3. Calculate the values of $a$ and $b$ from the denominators of the standard form equation.

4. Find the vertices and foci using $a$ and $c = \sqrt{a^2 + b^2}$, but stop before plugging in the final values.

5. Set up the equations for the asymptotes using the values of $a$ and $b$.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{x^2}{2} - \frac{(y+1)^2}{9} = 1$ Center: $(0, -1)$ Vertices: $(0, -1 \pm 3)$ Foci: $(0, -1 \pm \sqrt{11})$ Asymptotes: $y + 1 = \pm \frac{3}{\sqrt{2}}x$ The hyperbola opens up and down.

Q5. Identify the conic from the equation and put it in standard form.

Background

Topic: Identifying Conic Sections and Standard Forms

This question is testing your ability to recognize the type of conic section given an equation and to rewrite it in standard form. You may need to complete the square or rearrange terms to achieve this.

Key Terms and Formulas

• Standard form for a parabola: $4p(x-h) = (y-k)^2$ or $4p(y-k) = (x-h)^2$

• Standard form for an ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Standard form for a hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

Step-by-Step Guidance

1. Examine the given equation and determine which conic section it represents by looking at the degree and sign of the $x^2$ and $y^2$ terms.

2. If necessary, rearrange the equation and complete the square for $x$ and/or $y$ to rewrite it in standard form.

3. Identify the center, vertex, or other key features based on the standard form you obtain.

4. Set up the equation for the conic in standard form, but stop before simplifying all terms or plugging in the final values.

Try solving on your own before revealing the answer!

Final Answer:

Each equation is identified and rewritten in standard form: Ellipse: $\frac{(x-13)^2}{36} + \frac{(y+3)^2}{9} = 1$ Parabola: $(x-33) = -5(y+3)^2$ Hyperbola: $\frac{(x-3)^2}{6} - \frac{(y+3)^2}{9} = 1$ All conics are in standard form and their key features can be identified from these equations.