Lecture Notes: Grade 11 Parabola
Identifying a Parabola
- A parabola is identified by the equation of the form (y = x^2).
- The key feature distinguishing a parabola from an exponential graph:
- Parabola: (x^2) (x is the base)
- Exponential: (x) is an exponent.
Transformation of Parabolas
Vertical Shifts
- Rule: (y = x^2 + c)
- (c > 0): Shifts upward.
- (c < 0): Shifts downward.
- Example: (y = x^2 + 1) shifts 1 unit up from (0,0) to (0,1).
Horizontal Shifts
- Rule: (y = (x - h)^2)
- (h > 0): Shifts right.
- (h < 0): Shifts left.
- Think in opposites: (x - 2) shifts right.
- Example: ((x - 2)^2 + 1) shifts right 2 units, up 1 unit.
The Impact of the Leading Coefficient
- Coefficient (a) in (y = ax^2):
- (a > 0): "Happy" parabola (opens upward).
- (a < 0): "Sad" parabola (opens downward).
- Magnitude of (a): Larger (a) results in a narrower parabola.
Graph Sketching Practice
- Purpose: Understanding shifts and orientations.
- Not focused on exact intercept values, just shifts and direction.
Examples and Analysis
Example 1:
- Equation: (y = x^2 - 4)
- Type: Parabola with vertical shift down 4 units.
- Result: Happy parabola.
Example 2:
- Equation: (y = (x - 1)^2 - 4)
- Type: Horizontal shift right 1, vertical shift down 4.
- Result: Happy parabola.
Example 3:
- Equation: (y = -(x - 1)^2 + 4)
- Type: Horizontal shift right 1, vertical shift up 4.
- Result: Sad parabola.
Example 4:
- Equation: (y = (x + 4)^2 + 3)
- Type: Horizontal shift left 4, vertical shift up 3.
- Result: Happy parabola.
Example 5:
- Equation: (y = (x - 6)^2 + 2)
- Type: Horizontal shift right 6, vertical shift up 2.
- Result: Happy parabola.
Drawing a Parabola
Steps:
- X-Intercepts: Make (y = 0).
- Y-Intercept: Make (x = 0).
- Turning Point:
- Method 1: Average the x-intercepts.
- Method 2: Use the formula (-b/(2a)).
- Method 3: From turning point form.
Example Calculation
- Equation: (y = x^2 - 7x - 5)
- X-Intercepts: Find using quadratic formula.
- Y-Intercept: (y = -5).
- Turning Point:
- Calculated using (-b/(2a)) and substituting back into the equation.
Turning Point Form
- Form: (y = (x - h)^2 + k)
- Quick identification of turning point: ( (h, k) ).
- Use turning point formula when presented in this form.
In the next video, we will practice more examples and focus on detailed graphing of parabolas.