Unit 5 Retest - Algebra

Front

What is the vertex of a quadratic function in standard form?

Back

The vertex of a quadratic function in standard form, f(x) = ax^2 + bx + c, can be found using the formula: \( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \).

Front

How do you find the coordinates of the vertex for the function f(x) = x^2 - 12x + 8?

Back

The vertex can be found using the formula: \( x = -\frac{b}{2a} \). Here, a = 1 and b = -12, so x = 6. Plugging x back into the function gives f(6) = -28, so the vertex is (6, -28).

Front

What is the discriminant in the quadratic formula?

Back

The discriminant is given by the formula: \( D = b^2 - 4ac \). It determines the nature of the roots of the quadratic equation.

Front

What does a negative discriminant indicate about the roots of a quadratic equation?

Back

A negative discriminant indicates that the quadratic equation has two complex (imaginary) roots.

Front

What is the next step to solve the equation x^2 + 6x + 9 = -17 using complex numbers?

Back

The next step is to factor the left side to (x + 3)^2, which simplifies the equation.

Front

What is the vertex form of a quadratic function?

Back

The vertex form of a quadratic function is given by: \( f(x) = a(x - h)^2 + k \), where (h, k) is the vertex.

Front

How do you convert a quadratic function from standard form to vertex form?

Back

To convert from standard form to vertex form, complete the square on the quadratic expression.

Front

What is the maximum profit in the function P(x) = -x^2 + 60x - 100?

Back

The maximum profit can be found at the vertex of the parabola represented by the function. The vertex occurs at x = 30, giving P(30) = 800.

Front

What is the significance of the coefficient 'a' in a quadratic function?

Back

The coefficient 'a' determines the direction of the parabola: if 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

Front

What is the formula for the quadratic formula?

Back

The quadratic formula is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).