Solved Exercises on Parabolas (Equations and Graphs) in Algebra 2

Master parabolas: equations, graphs, vertices, foci, directrices, and problem-solving techniques through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Vertex Form Analysis
Exercise 1
For the parabola y = 2(x - 3)² + 4, find the vertex, focus, directrix, and axis of symmetry.
Definition:

Parabola: The set of all points equidistant from a fixed point (focus) and a fixed line (directrix)

Vertex Form: y = a(x - h)² + k, where (h, k) is the vertex

\(y = a(x - h)^2 + k\)
Vertex Form of Vertical Parabola
Solution Method:
  1. Identify parameters a, h, k from vertex form
  2. Calculate focal distance p = 1/(4a)
  3. Determine focus and directrix locations
  4. Identify axis of symmetry
Given
y=2(x-3)²+4
Parameters
a=2, h=3, k=4
Result
Vertex:(3,4)
Step 1: Identify parameters from vertex form

y = 2(x - 3)² + 4

Comparing to y = a(x - h)² + k:

a = 2, h = 3, k = 4

Step 2: Find the vertex

Vertex: (h, k) = (3, 4)

Step 3: Calculate focal distance

p = 1/(4a) = 1/(4 × 2) = 1/8

Step 4: Determine focus and directrix

Since a > 0, parabola opens upward

Focus: (h, k + p) = (3, 4 + 1/8) = (3, 33/8)

Directrix: y = k - p = 4 - 1/8 = 31/8

Step 5: Identify axis of symmetry

Axis of symmetry: x = h = 3

Vertex: (3,4), Focus: (3,33/8), Directrix: y = 31/8, Axis: x = 3
Final answer:

Vertex: (3, 4), Focus: (3, 33/8), Directrix: y = 31/8, Axis of symmetry: x = 3

Applied rules:

• Parameter p: p = 1/(4a) relates coefficient to focus distance

• Focus location: Inside parabola, distance p from vertex

• Directrix: Perpendicular to axis of symmetry, distance p from vertex

Tip 1: The sign of 'a' determines the direction the parabola opens.
Tip 2: Focus and directrix are equidistant from the vertex.
2 Horizontal Parabola
Exercise 2
For the parabola (y - 2)² = 8(x + 1), find the vertex, focus, and directrix.
Definition:

Horizontal Parabola: Opens left or right, with axis parallel to x-axis

Standard Form: (y - k)² = 4p(x - h), where (h, k) is the vertex

\((y - k)^2 = 4p(x - h)\)
Standard Form of Horizontal Parabola
Given
(y-2)²=8(x+1)
Standard
(y-k)²=4p(x-h)
Result
Vertex:(-1,2)
Step 1: Compare to standard form

(y - 2)² = 8(x + 1)

(y - k)² = 4p(x - h)

Step 2: Identify parameters

k = 2 (from y - 2)

h = -1 (from x - (-1) = x + 1)

4p = 8, so p = 2

Step 3: Find the vertex

Vertex: (h, k) = (-1, 2)

Step 4: Determine focus and directrix

Since p > 0, parabola opens to the right

Focus: (h + p, k) = (-1 + 2, 2) = (1, 2)

Directrix: x = h - p = -1 - 2 = -3

Step 5: Identify axis of symmetry

Axis of symmetry: y = k = 2

Vertex: (-1,2), Focus: (1,2), Directrix: x = -3, Axis: y = 2
Final answer:

Vertex: (-1, 2), Focus: (1, 2), Directrix: x = -3, Axis of symmetry: y = 2

Applied rules:

• Horizontal orientation: y² term is isolated, parabola opens horizontally

• Focus location: Inside parabola along axis of symmetry

• Directrix: Perpendicular to axis of symmetry, distance p from vertex

Tip 1: Horizontal parabolas have y² as the isolated term.
Tip 2: For horizontal parabolas, focus moves horizontally from vertex.
3 Graphing Parabola
Exercise 3
Graph the parabola y = -(x + 2)² + 5 and identify its domain, range, and intercepts.
Definition:

Domain: All possible x-values for which the function is defined

Range: All possible y-values the function can take

Intercepts: Points where the graph crosses the axes

\(y = -(x + 2)^2 + 5\)
Vertex Form (a < 0)
Given
y=-(x+2)²+5
Parameters
a=-1, h=-2, k=5
Result
Vertex:(-2,5)
Step 1: Identify parameters from vertex form

y = -(x + 2)² + 5

Comparing to y = a(x - h)² + k:

a = -1, h = -2, k = 5

Step 2: Find the vertex and direction

Vertex: (h, k) = (-2, 5)

Since a = -1 < 0, parabola opens downward

Step 3: Determine domain and range

Domain: All real numbers (since parabola extends infinitely in x-direction)

Range: y ≤ 5 (since vertex is maximum point and parabola opens downward)

Step 4: Find y-intercept

Set x = 0: y = -(0 + 2)² + 5 = -4 + 5 = 1

y-intercept: (0, 1)

Step 5: Find x-intercepts

Set y = 0: 0 = -(x + 2)² + 5

(x + 2)² = 5

x + 2 = ±√5

x = -2 ± √5

x-intercepts: (-2 + √5, 0) and (-2 - √5, 0)

Domain: (-∞, ∞), Range: (-∞, 5], Intercepts: (0,1), (-2±√5,0)
Final answer:

Domain: (-∞, ∞), Range: (-∞, 5], y-intercept: (0, 1), x-intercepts: (-2 + √5, 0) and (-2 - √5, 0)

Applied rules:

• Domain of parabola: Always all real numbers

• Range determination: Depends on vertex and direction of opening

• Intercept calculation: Set opposite variable to zero and solve

Tip 1: For finding x-intercepts, set y = 0 and solve the quadratic equation.
Tip 2: Range is determined by the vertex when a < 0 (maximum) or a > 0 (minimum).
Key Formulas and Properties
\(y = a(x - h)^2 + k\)
Vertical Parabola (Vertex Form)
\((y - k)^2 = 4p(x - h)\)
Horizontal Parabola
\(p = \frac{1}{4a} \text{ or } 4p = \frac{1}{a}\)
Focal Parameter
Vertical
Opens Up/Down
Focus: (h, k±p), Directrix: y=k∓p
Horizontal
Opens Left/Right
Focus: (h±p, k), Directrix: x=h∓p
Direction
Coefficient Sign
a > 0: opens up/right, a < 0: opens down/left
Key Definitions:

Parabola: Set of points equidistant from focus and directrix

Vertex: Extreme point of the parabola

Focus: Fixed point inside the parabola

Directrix: Fixed line outside the parabola

Axis of Symmetry: Line through vertex perpendicular to directrix

Problem-Solving Strategy:
  1. Identify orientation: Determine if parabola opens vertically or horizontally
  2. Convert to standard form: Rewrite equation in vertex form
  3. Extract parameters: Identify a, h, k, and calculate p
  4. Apply formulas: Calculate focus, directrix, and other features
  5. Verify relationships: Check that all points satisfy the definition
Common Errors: Sign errors in vertex coordinates, incorrect focal distance calculation, confusion between vertical and horizontal orientations.
Exam Tips: Memorize both forms, practice converting between forms, understand parameter relationships.
Solution: Exercises 4 to 5
4 Real-world Application
Exercise 4
A satellite dish has a parabolic cross-section. The dish is 8 feet wide and 2 feet deep at its center. Find the equation of the parabola and determine where the receiver should be placed.
Definition:

Parabolic Reflector: Uses the property that all incoming parallel rays reflect through the focus

Setup
Width=8ft, Depth=2ft
Equation
y=ax²
Result
Focus at (0,1/2)
Step 1: Set up coordinate system

Place vertex at origin, with parabola opening upward

Parabola passes through points (±4, 2) since width is 8 feet and depth is 2 feet

Step 2: Find the equation

Using vertex form: y = ax²

Substitute point (4, 2): 2 = a(4)² = 16a

Therefore: a = 2/16 = 1/8

Equation: y = (1/8)x²

Step 3: Calculate focal distance

For y = ax², p = 1/(4a) = 1/(4 × 1/8) = 1/(1/2) = 2

Step 4: Determine focus location

Since vertex is at (0, 0) and parabola opens upward:

Focus: (0, p) = (0, 2)

Step 5: Interpret result

The receiver should be placed 2 feet above the vertex of the dish

Equation: y = (1/8)x², Receiver placement: 2 feet above vertex
Final answer:

Parabola equation: y = (1/8)x², Receiver should be placed 2 feet above the vertex

Applied rules:

• Modeling: Translate physical situation to mathematical equation

• Focus property: Parallel rays reflect through focus in parabolic reflectors

• Coordinate geometry: Use geometric constraints to find parameters

Tip 1: Always place the vertex at the origin when possible for easier calculations.
Tip 2: The focus is where signals collect in parabolic reflectors.
5 Converting to Standard Form
Exercise 5
Convert the equation y = x² - 6x + 8 to vertex form and identify all key features of the parabola.
Definition:

Completing the Square: Technique to convert general form to vertex form

Original
y=x²-6x+8
Complete square
y=(x-3)²-1
Result
Vertex:(3,-1)
Step 1: Start with the general form

y = x² - 6x + 8

Step 2: Group x terms and factor out coefficient

y = (x² - 6x) + 8

(Coefficient of x² is already 1)

Step 3: Complete the square

Take half of coefficient of x: -6/2 = -3

Square it: (-3)² = 9

Add and subtract 9 inside the parentheses:

y = (x² - 6x + 9 - 9) + 8

y = (x² - 6x + 9) - 9 + 8

Step 4: Factor perfect square and simplify

y = (x - 3)² - 1

Step 5: Identify key features

Vertex form: y = (x - 3)² + (-1)

Vertex: (3, -1)

a = 1 > 0, so parabola opens upward

p = 1/(4a) = 1/4

Focus: (3, -1 + 1/4) = (3, -3/4)

Directrix: y = -1 - 1/4 = -5/4

Vertex form: y = (x - 3)² - 1, Vertex: (3,-1), Focus: (3,-3/4)
Final answer:

Vertex form: y = (x - 3)² - 1, Vertex: (3, -1), Focus: (3, -3/4), Directrix: y = -5/4

Applied rules:

• Completing the square: Add (coefficient/2)² to both sides

• Perfect square trinomial: x² + bx + (b/2)² = (x + b/2)²

• Vertex form recognition: Identify parameters from y = a(x - h)² + k

Tip 1: Always add and subtract the same value when completing the square.
Tip 2: Factor out the leading coefficient if it's not 1 before completing the square.
Comprehensive Guide: Parabolas (Equations and Graphs)
\(y = a(x - h)^2 + k \text{ or } x = a(y - k)^2 + h\)
Vertex Form of Parabolas
\((y - k)^2 = 4p(x - h) \text{ or } (x - h)^2 = 4p(y - k)\)
Standard Form of Parabolas
Key definitions:

Parabola: The set of all points equidistant from a fixed point (focus) and a fixed line (directrix)

Vertex: The extreme point of the parabola, located halfway between focus and directrix

Focus: The fixed point inside the parabola that defines its shape

Directrix: The fixed line outside the parabola that defines its shape

Axis of Symmetry: The line through the vertex perpendicular to the directrix

Complete methodology:
  1. Identify orientation: Determine if parabola opens vertically or horizontally
  2. Convert to standard form: Rewrite equation in vertex or standard form
  3. Extract parameters: Identify a, h, k, and calculate p = 1/(4a)
  4. Calculate features: Find vertex, focus, directrix, axis of symmetry
  5. Graph the parabola: Plot key points and sketch the curve
Tip 1: The sign of 'a' determines the direction of opening: positive = upward/right, negative = downward/left.
Tip 2: The larger the absolute value of 'a', the narrower the parabola.
Tip 3: Focus is always inside the parabola, and directrix is always outside.
Tip 4: The distance from vertex to focus equals the distance from vertex to directrix.
Common errors: Sign errors in vertex coordinates, incorrect focal distance calculation, confusion between vertical and horizontal orientations, calculation errors in completing the square.
Exam preparation: Memorize both vertex and standard forms, practice converting between forms, work on real-world applications, master completing the square technique.
Essential properties to know:

• Vertex form reveals vertex (h, k) directly

• Focal distance: p = 1/(4a) for y = a(x - h)² + k

• Focus and directrix are equidistant from vertex

• Axis of symmetry passes through vertex and focus

Visual Understanding: Parabola Properties
Exercise 6: Parabola Comparison
Compare different parabolas:
• y = x² (a = 1, opens upward)
• y = -x² (a = -1, opens downward)
• y = 2x² (a = 2, narrow parabola)
• y = (1/2)x² (a = 1/2, wide parabola)

Analysis: The visualization shows how different coefficients affect the shape and direction of parabolas.

  • Positive 'a' values create upward-opening parabolas
  • Negative 'a' values create downward-opening parabolas
  • Larger absolute values of 'a' create narrower parabolas
  • Smaller absolute values of 'a' create wider parabolas

Questions & Answers

Question: How do I know if a parabola opens vertically or horizontally?

Answer: Look at which variable is squared:

  • Vertical parabolas: y is expressed in terms of x² (like y = ax² + bx + c)
  • Horizontal parabolas: x is expressed in terms of y² (like x = ay² + by + c)

In vertex form:
• y = a(x - h)² + k: opens vertically (up/down)
• x = a(y - k)² + h: opens horizontally (left/right)

In standard form:
• (y - k)² = 4p(x - h): opens horizontally
• (x - h)² = 4p(y - k): opens vertically

The key is identifying which variable appears by itself on one side of the equation.

Question: What's the relationship between the focus and directrix in a parabola?

Answer: The relationship is fundamental to the definition of a parabola:

  • Equal distances: Any point on the parabola is equidistant to the focus and directrix
  • Opposite sides: The focus is inside the parabola, while the directrix is outside
  • Vertex midpoint: The vertex is equidistant from both focus and directrix
  • Perpendicular: The directrix is perpendicular to the axis of symmetry

If P is any point on the parabola, F is the focus, and D is the directrix, then distance(P,F) = distance(P,D).

This property is what gives parabolas their unique reflective characteristics.

Question: What are some real-world applications of parabolas? Why is it important to learn this?

Answer: Parabolas have numerous practical applications:

  • Satellite dishes: Collect signals at the focus
  • Car headlights: Reflect light in parallel beams
  • Projectile motion: Path of thrown objects follows a parabolic trajectory
  • Architecture: Bridge design, arches, and structural elements
  • Optics: Lenses and mirrors in telescopes and cameras

Learning parabolas develops understanding of curved geometric shapes and their properties. This knowledge is fundamental to physics, engineering, astronomy, and architecture. The mathematical techniques used (completing the square, coordinate geometry) are essential for higher mathematics.

Understanding parabolas helps in modeling real-world phenomena and solving practical problems in science and engineering.

Question: I sometimes struggle with completing the square for parabolas. Any tips?

Answer: Here are strategies for completing the square in parabolas:

  1. Factor out coefficient: If a ≠ 1, factor it out from x² and x terms
  2. Take half and square: Take half of the linear coefficient, then square it
  3. Balance the equation: Whatever you add inside parentheses, multiply by the factored coefficient when moving outside
  4. Write as perfect square: Express as (x ± number)²
  5. Simplify constants: Combine remaining terms

For example, with y = 2x² + 8x + 5:
1. Factor: y = 2(x² + 4x) + 5
2. Half of 4 is 2, squared is 4: y = 2(x² + 4x + 4 - 4) + 5
3. Balance: y = 2((x + 2)² - 4) + 5
4. Distribute: y = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3

Always verify by expanding your completed square to ensure it matches the original expression.

Question: How do I find the domain and range of a parabola?

Answer: The domain and range depend on the parabola's orientation:

  • Domain: For all parabolas, domain is (-∞, ∞) since they extend infinitely in the input direction
  • Vertical parabolas:
    • Opens upward: range is [k, ∞) where k is the y-coordinate of the vertex
    • Opens downward: range is (-∞, k] where k is the y-coordinate of the vertex
  • Horizontal parabolas:
    • Opens rightward: domain is [h, ∞) where h is the x-coordinate of the vertex
    • Opens leftward: domain is (-∞, h] where h is the x-coordinate of the vertex

For example, y = (x - 3)² + 2 has vertex (3, 2) and opens upward, so domain is (-∞, ∞) and range is [2, ∞).

The key insight is that the vertex represents the extreme value of the dependent variable.