Overview
This lecture explains how to find the vertex of a parabola for quadratic equations given in standard form, vertex form, and factored form.
Finding the Vertex in Standard Form
- Standard form: ( y = ax^2 + bx + c ).
- The x-coordinate of the vertex is ( x = -\frac{b}{2a} ).
- Substitute x back into the equation to find the y-coordinate: ( y = a x^2 + b x + c ).
- Example: For ( y = x^2 - 4x + 3 ), vertex at (2, -1).
- Example: For ( y = -2x^2 + 8x - 5 ), vertex at (2, 3).
- Handle cases where ( b = 0 ) or x-coordinate is a fraction.
Finding the Vertex in Vertex Form
- Vertex form: ( y = a(x - h)^2 + k ).
- Vertex is at ( (h, k) ), where h is the opposite sign of the term inside parentheses.
- Example: ( y = 3(x - 2)^2 + 4 ), vertex is (2, 4).
- For ( y = 5(x + 4)^2 - 5 ), vertex is (-4, -5).
- Do not change the sign of k.
Finding the Vertex in Factored Form
- Factored form: ( y = a(x - r_1)(x - r_2) ).
- X-intercepts are ( r_1 ) and ( r_2 ).
- X-coordinate of the vertex is the midpoint: ( x = \frac{r_1 + r_2}{2} ).
- Find y-coordinate by substituting x back into the original equation.
- Alternatively, expand to standard form and use ( x = -\frac{b}{2a} ).
- Both methods give the same result.
Key Terms & Definitions
- Standard Form — ( y = ax^2 + bx + c )
- Vertex Form — ( y = a(x - h)^2 + k )
- Factored Form — ( y = a(x - r_1)(x - r_2) )
- Vertex — The turning point of the parabola, coordinates ((x, y)).
Action Items / Next Steps
- Practice finding the vertex for quadratic equations in all three forms.
- Try the examples provided in the lecture for further understanding.