1. Let's start by understanding the vertex form of a quadratic function. The vertex form is given by: $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex of the parabola, and $a$ determines the direction and width of the parabola. 2. Important rules: - If $a > 0$, the parabola opens upwards. - If $a < 0$, the parabola opens downwards. - The vertex $(h,k)$ is the highest or lowest point on the graph. 3. To convert a quadratic function from standard form $y = ax^2 + bx + c$ to vertex form, use the method of completing the square: Start with $y = ax^2 + bx + c$ Divide all terms by $a$ (if $a \neq 1$): $$y = a\left(x^2 + \frac{b}{a}x\right) + c$$ 4. Complete the square inside the parentheses: Take half of $\frac{b}{a}$, square it, and add and subtract it inside the parentheses: $$y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$ 5. Rewrite the perfect square trinomial and simplify: $$y = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c$$ 6. Distribute $a$ and combine constants: $$y = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c$$ Simplify the constant term: $$- a\left(\frac{b}{2a}\right)^2 + c = - \frac{b^2}{4a} + c$$ 7. So the vertex form is: $$y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$$ where the vertex is at $\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$. 8. Example: Convert $y = 2x^2 - 8x + 3$ to vertex form. Start: $$y = 2x^2 - 8x + 3$$ Divide by 2: $$y = 2\left(x^2 - 4x\right) + 3$$ Complete the square: Half of $-4$ is $-2$, square is 4: $$y = 2\left(x^2 - 4x + 4 - 4\right) + 3$$ Rewrite: $$y = 2\left((x - 2)^2 - 4\right) + 3$$ Distribute 2: $$y = 2(x - 2)^2 - 8 + 3$$ Simplify constants: $$y = 2(x - 2)^2 - 5$$ Vertex is at $(2, -5)$. 9. Practice predicting questions: - Convert quadratic functions from standard to vertex form. - Identify vertex, axis of symmetry, and direction of opening. - Graph quadratic functions using vertex form. - Solve quadratic equations by completing the square. 10. Bonus tip: Remember the axis of symmetry is $x = h$ from the vertex form. Good luck on your test!