CH 12 Review: Practice Quiz Questions

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a is not zero. This form is essential for solving and analyzing quadratic functions.

A quadratic function always graphs as a parabola. The parabola opens upward if the coefficient of x^2 is positive and downward if it is negative.

The axis of symmetry for a quadratic function in standard form is given by x = -b/(2a). This line splits the parabola into two mirror-image halves and is key to understanding the graph's structure.

The discriminant, calculated as b^2 - 4ac, determines whether the roots are real or complex. A positive discriminant gives two distinct real roots, zero gives a repeated real root, and a negative discriminant indicates complex roots.

The expression x^2 + 5x + 6 factors neatly into (x + 2)(x + 3) because 2 and 3 add up to 5 and multiply to 6. This is a typical example of factoring a simple quadratic.

Completing the square transforms a quadratic function into vertex form, making it easier to identify the vertex. The process rearranges the quadratic into a perfect square plus a constant term.

Using the quadratic formula x = [-b ± √(b^2 - 4ac)]/(2a) with a = 2, b = -4, and c = -6 gives a discriminant of √64 which simplifies to 8. This results in the solutions x = (4 ± 8)/4, or x = 3 and x = -1.

By searching for two numbers that multiply to 18 (3 multiplied by 6) and add to 11, you find 2 and 9. This allows the expression to be factored as (3x + 2)(x + 3) after grouping.

The vertex of a quadratic function in standard form is found using x = -b/(2a). For the function 2x^2 - 8x + 3, this gives x = 2, and substituting back yields y = -5, so the vertex is (2, -5).

A negative discriminant, found by computing 4^2 - 4(1)(5) = -4, indicates that the quadratic does not have real solutions. Instead, it has two complex conjugate roots.

The expression (x - 3)^2 indicates a horizontal translation 3 units to the right compared to x^2. This shift is a basic transformation property of quadratic functions.

To complete the square, half of 6 is 3 and its square is 9. Rewriting the expression as (x + 3)^2 and adjusting for the constant gives (x + 3)^2 - 4.

The maximum height of a projectile is found at the vertex of its quadratic height function. Using t = -b/(2a) for h(t) results in t = 1 second, when the height is maximized.

The coefficient 'a' controls both the concavity and the width of the parabola. A larger absolute value of a makes the parabola narrower, while its sign determines whether it opens upward or downward.

The equation factors neatly as (x - 2)(x - 5) = 0. Setting each factor equal to zero yields the solutions x = 2 and x = 5.

After factoring the inequality to (x - 2)(x - 3) < 0, testing shows that the expression is negative when x lies between the roots 2 and 3. This interval is the solution to the inequality.

Starting with the vertex form f(x) = a(x - 2)^2 - 3 and substituting the point (4, 5) allows solving for a. The calculation reveals that a = 2, resulting in the quadratic function f(x) = 2(x - 2)^2 - 3.

A quadratic has one real repeated root when its discriminant is zero. For the equation x^2 + kx + 9, setting k^2 - 36 = 0 leads to k^2 = 36, so k must be either 6 or -6.

Using the perimeter formula 2(width + length) = 200 along with the constraint length = width + 10, we form the equation 2(x + x + 10) = 200. Solving this equation gives a width of 45 meters and a length of 55 meters.

For the parabola to open upward, the coefficient a must be positive. Also, the vertex is located at x = -b/(2a), so setting this equal to 4 yields b = -8a, ensuring the stated properties of the quadratic function.