Quadratic Functions and Equations Practice Quiz

The standard form of a quadratic equation is written as ax^2 + bx + c = 0, where a ≠ 0. This form helps in identifying the coefficients and applying solution methods effectively.

The vertex of the parabola is the point where the quadratic function reaches its highest or lowest value. While the axis of symmetry divides the parabola into mirror images, the vertex actually represents the extreme value.

Factorization, completing the square, and the quadratic formula are the most common methods for solving quadratic equations. The substitution method is not a standard technique for solving quadratics.

The expression b² - 4ac is known as the discriminant and it determines the nature of the quadratic equation's roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real repeated root, and a negative discriminant indicates two complex roots.

The y-intercept of a quadratic function is found by setting x = 0, which yields f(0) = c. Therefore, the constant term c represents the y-intercept.

The discriminant is computed as b² - 4ac. For the equation 2x² - 4x - 6 = 0, we have (-4)² - 4(2)(-6) = 16 + 48 = 64, which indicates the presence of two distinct real roots.

Rewriting x² + 6x + 5 = 0 by completing the square gives (x + 3)² = 4, leading to solutions x = -3 ± 2. This results in x = -1 or x = -5, and -1 is one of the valid solutions.

The vertex form, y = a(x - h)² + k, clearly identifies the vertex (h, k) of the parabola and the coefficient a which affects the graph's stretch. The other forms do not explicitly reveal the vertex's location.

The function is given in vertex form with the vertex at (2, 4), meaning the vertical line x = 2 is the axis of symmetry. This line divides the parabola into two mirror-image halves.

A discriminant of 0 indicates that the quadratic equation touches the x-axis at a single point. This means the equation has one real repeated root, sometimes referred to as a double root.

The coefficient 'a' in a quadratic function influences whether the parabola opens upward or downward and also affects its width. A larger absolute value of a makes the parabola narrower, while a smaller value makes it wider.

The constant term 'c' in the quadratic function represents the y-intercept. Changing c results in a vertical shift of the graph up or down, without affecting its horizontal position or shape.

Factoring x² - 5x + 6 gives (x - 2)(x - 3) = 0, leading to solutions x = 2 and x = 3. The sum of these solutions, 2 + 3, is 5, which also aligns with Vieta's formula.

A parabola opens downward when the coefficient of x² is negative. In the equation y = -2x² + 4x - 1, the coefficient -2 confirms that the graph opens downward.

In the vertex form, y = (x-3)² + 2 indicates a horizontal shift 3 units to the right and a vertical shift 2 units upward relative to the basic parabola y = x². This transformation moves the vertex from (0, 0) to (3, 2).

For the parabola to touch the x-axis, its discriminant must be zero. Setting b² - 4ac = (-8)² - 4(2)(k) = 64 - 8k equal to 0 results in k = 8, ensuring a single repeated real root.

The vertex formula x = -b/(2a) is a direct and efficient method for finding the x-coordinate of the vertex from the standard form of a quadratic function. It avoids the more time-consuming processes of factoring or plotting numerous points.

The axis of symmetry for a quadratic equation in standard form is given by x = -b/(2a). Substituting a = 5 and b = 20 yields x = -20/(10) = -2.

The vertex form of the quadratic function reveals the vertex, which in this case is (-1, 7). Because the coefficient -3 is negative, the parabola opens downward and the vertex represents the maximum value, which is 7.

Applying the quadratic formula with a = 3, b = -2, and c = -5 produces a discriminant of 64. This yields the solutions x = (2 ± 8) / 6, simplifying to x = 5/3 and x = -1.