Characteristics Of Quadratic Functions Quiz

The graph of any quadratic function is always a parabola. This is due to the presence of the x² term, which gives the graph its curved shape. Unlike straight lines or other curves, quadratic functions form a U-shaped or inverted U-shape graph, depending on the sign of the leading coefficient. A positive coefficient creates a U-shape, while a negative coefficient forms an inverted U.

The direction of the parabola for the equation y = –2x² + 4x – 1 is determined by the leading coefficient. Since the coefficient of x² is negative (-2), the parabola opens downward. This means the graph starts high at the vertex and curves down as x moves away from the vertex. The negative coefficient indicates a maximum point at the vertex.

To find the y-intercept, substitute x = 0 into the quadratic equation y = 3x² – 12x + 7. This gives y = 7, so the y-intercept is the point (0, 7). The y-intercept occurs when x equals zero, and this value is the constant term (7) in the quadratic equation. The other options do not represent the correct intercept.

To find the vertex of the quadratic y = x² – 6x + 11, use the formula for the x-coordinate of the vertex: x = -b/(2a). Here, a = 1 and b = -6, so the x-coordinate is 3. Substituting x = 3 into the equation gives the y-coordinate as 2. Therefore, the vertex is (3, 2), which is the minimum point of the parabola.

The axis of symmetry for a quadratic function is calculated using x = -b/(2a). For the quadratic y = –4x² + 20x – 13, a = -4 and b = 20. Using the formula, x = 2.5. The axis of symmetry is the vertical line x = 2.5, which passes through the vertex. This line divides the parabola into two symmetric halves.

The quadratic function y = –x² + 8x – 2 has a negative leading coefficient (-1), which means the parabola opens downward. A downward-opening parabola has a maximum point at the vertex. To find the maximum, calculate the vertex using x = -b/(2a). This gives the vertex as (4, 14), which represents the maximum value of the function.

To find the x-intercepts of y = 2x² – 8x + 5, solve the equation using the quadratic formula. The discriminant is positive, indicating two real roots. The roots are x = 2 ± √6 / 2, which are irrational. These values are approximately x = 2 ± 0.612. Therefore, the quadratic has two distinct real x-intercepts, as confirmed by the discriminant.

The equation y = –(x + 4)² + 9 is in vertex form, where the vertex is given by the coordinates (h, k). Here, h = -4 and k = 9, so the vertex is (–4, 9). Since the coefficient of the squared term is negative, the parabola opens downward. This form allows easy identification of the vertex and the direction of the parabola.

The maximum height of y = –3x² + 12x – 7 occurs at the vertex. To find the x-coordinate of the vertex, use x = -b/(2a). Substituting the values gives x = 2. Substituting this into the equation gives the maximum height as 5 meters, which is the y-coordinate at the vertex. Since the parabola opens downward, this is the highest point.

The domain of any quadratic function is all real numbers (x ∈ ℝ) because quadratic functions can take any value of x. The range depends on the vertex. For y = 2x² + 8x + 3, the minimum value occurs at x = -2, and the y-value at the vertex is -5. Therefore, the range is y ≥ -5, indicating all y-values are greater than or equal to -5.

The minimum value of y = 4x² – 16x + 10 occurs at the vertex. Using the formula x = -b/(2a), the x-coordinate of the vertex is 2. Substituting this into the equation gives the minimum value as –6. Since the parabola opens upward (positive leading coefficient), the vertex represents the lowest point on the graph.

A positive leading coefficient indicates that the parabola opens upward. In this case, as x increases or decreases from the vertex, y increases, making the vertex a minimum point. The graph's arms extend upwards, and the function has no maximum. A negative leading coefficient would open downward, with a maximum at the vertex.

The x-intercepts of y = x² – 9 can be found by setting y = 0. This gives the equation x² – 9 = 0, which factors as (x - 3)(x + 3) = 0. Therefore, the x-intercepts are x = 3 and x = -3. These values are the points where the graph crosses the x-axis, and they are symmetric about the origin.

The axis of symmetry for y = x² + 2x – 3 is calculated using x = -b/(2a). Here, a = 1 and b = 2, so the axis of symmetry is x = -1. This is the vertical line that passes through the vertex of the parabola, ensuring symmetry on both sides. The vertex of this parabola is located at (–1, –4).

The quadratic function y = x² + 1 has no real roots because the discriminant is negative. The discriminant for this function is b² - 4ac = 0 - 4 = -4, which is less than 0. A negative discriminant means the quadratic does not intersect the x-axis, and the roots are complex numbers. Other functions listed have real roots based on their discriminants.