(Alg.2) 4.5 (Flashcard) Converting and Transformations.

The standard form of a quadratic equation is given by $$y = ax^2 + bx + c$$ , where a, b, and c are constants and a ≠ 0.

To convert from vertex form $$y = a(x-h)^2 + k$$ to standard form, expand the equation: $$y = a(x^2 - 2hx + h^2) + k$$ .

A parabola is reflected over the x-axis when the coefficient of $$x^2$$ is negative, changing the direction of the opening.

The vertex form of a quadratic equation is given by $$y = a(x-h)^2 + k$$ , where (h, k) is the vertex of the parabola.

For a quadratic in standard form $$y = ax^2 + bx + c$$ , the vertex can be found using the formula $$h = -\frac{b}{2a}$$ and then substituting h back into the equation to find k.

To convert from factored form $$y = (x - p)(x - q)$$ to standard form, expand the equation: $$y = x^2 - (p+q)x + pq$$ .

Transformation refers to changes made to the graph of a function, including translations, reflections, stretches, and compressions.

Changing the value of 'a' affects the width and direction of the parabola: if |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider; if a < 0, it opens downwards.

The direction of a parabola can be identified from the coefficient of $$x^2$$ : if it is positive, the parabola opens upwards; if negative, it opens downwards.

The vertex represents the maximum or minimum point of the parabola, depending on the direction it opens.