Algebra 2 Midterm Review

The vertex is the highest or lowest point of the parabola, represented as a coordinate (h, k) in the vertex form of a quadratic equation: y = a(x-h)^2 + k.

The maximum value occurs at the vertex of the parabola when it opens downwards. It is the y-coordinate of the vertex.

To find the roots, factor the equation: $$(x - 7)(x + 7) = 0$$ . The roots are $$x = 7$$ and $$x = -7$$ .

The decreasing intervals are the ranges of x-values where the function's output (y-value) is decreasing. For example, if the vertex is at (0, 2) and the parabola opens upwards, the decreasing interval is from (-∞, 0].

The x-intercepts can be found by setting the quadratic equation to zero and solving for x. For example, if the x-intercepts are x = -1 and x = 5, the equation can be written as $$(x + 1)(x - 5) = 0$$ .

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

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It can be found using the formula $$x = -\frac{b}{2a}$$ .

A quadratic function opens upwards if the coefficient of $$x^2$$ (a) is positive, and it opens downwards if a is negative.

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

To complete the square, rearrange the equation to isolate the constant term, then add and subtract the square of half the coefficient of x. For example, for $$x^2 + 6x$$ , add and subtract $$9$$ to get $$(x + 3)^2 - 9$$ .