Quadratic Functions: Factored Form Flashcards

The graph of a quadratic function is a parabola. The x-intercepts of the function are the points where the graph crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x. In this case, the function is (x + 3)(x + 1) = 0. To make this equation true, one or both of the factors must be zero. Therefore, x + 3 = 0 or x + 1 = 0. Solving for x gives us x = -3 and x = -1. These are the x-intercepts of the function. Because the function is negative, the parabola opens downwards.

The x-intercept of a function is the point where the graph crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercepts of f(x) = (x - 4)(x + 1), we set the function equal to zero and solve for x. That is, (x - 4)(x + 1) = 0. To make the equation true, either of the factors must be zero. So, x - 4 = 0 or x + 1 = 0. Solving for x, we get x = 4 and x = -1. Therefore, the x-intercepts of the function are (4, 0) and (-1, 0).

The axis of symmetry is a vertical line that divides the parabola into two equal halves. The x-coordinate of the axis of symmetry is equal to the average of the x-coordinates of the x-intercepts. In this case, the x-intercepts of f(x) = -(x + 9)(x - 21) are x = -9 and x = 21. The average is (-9 + 21) / 2 = 12 / 2 = 6. Therefore, the axis of symmetry is x = 6.

The range of a function is the set of all possible y-values. The minimum value of the function is the y-coordinate of the vertex, and the maximum value is positive infinity. Because the function is negative, the parabola opens downwards. Therefore, the range of the function f(x) = -(x + 5)(x + 1) is all real numbers less than or equal to the y-coordinate of the vertex.

The x-intercept is a point on the graph where the function crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept of f(x) = (x + 6)(x - 3), set the function equal to zero and solve for x. That is, (x + 6)(x - 3) = 0. For this equation to be true, one or both of the factors must be zero. This means x + 6 = 0 or x - 3 = 0. Solving for x, we get x = -6 and x = 3. The x-intercepts of the function are (-6, 0) and (3, 0).

The range of a function is the set of all possible y-values. The minimum value of the function is the y-coordinate of the vertex, and the maximum value is positive infinity. Because the function is positive, the parabola opens upwards. Therefore, the range of the function f(x) = (x − 4)(x − 2) is all real numbers greater than or equal to the y-coordinate of the vertex.

The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept of f(x) = (x - 8)(x + 3), set x to zero and evaluate the function. f(0) = (0 - 8)(0 + 3) = -24. So, the y-intercept is (0, -24).

The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept of f(x) = (x - 6)(x - 2), set x to zero and evaluate the function. f(0) = (0 - 6)(0 - 2) = 12. Therefore, the y-intercept is (0, 12).

The midpoint of the x-intercepts is the point on the x-axis that is equidistant from the two x-intercepts. To find the midpoint, we calculate the average of the x-coordinates of the x-intercepts. The x-intercepts of f(x) = (x - 2)(x - 4) are x = 2 and x = 4. The average of these values is (2 + 4) / 2 = 3. Therefore, the midpoint of the x-intercepts is (3, 0).

A quadratic function can have one, two, or no x-intercepts. To find the x-intercept, set the function equal to zero and solve for x. In this case, f(x) = x^2 + 12x + 36 = (x + 6)^2. To make the equation true, (x + 6)^2 = 0. Solving for x, we get x = -6. The x-intercept of the function is (-6, 0). This function has only one x-intercept.

The graph of a quadratic function is a parabola. The function is positive when the graph is above the x-axis and decreasing when the slope of the graph is negative. The x-intercepts of the function f(x) = (x − 3)(x + 1) are x = 3 and x = −1. The function is decreasing in the interval from x = −1 to x = 3. However, the function is positive only in the interval between the x-intercepts. Therefore, the function is positive and decreasing for all x-values between −1 and 3.

The graph of a quadratic function is a parabola. The x-intercepts of the function are the points where the graph crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x. Given that the function has two x-intercepts at (0, 0) and (4, 0), it means that when x = 0 and x = 4, the function equals zero. We can express this as (x - 0)(x - 4) = x(x - 4). The graph A represents this function because its x-intercepts are at x = 0 and x = 4.

The graph of a quadratic function is a parabola. The x-intercepts of the function are the points where the graph crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x. In this case, the function is (x - 1)(x + 4) = 0. To make the equation true, one or both of the factors must be zero. Therefore, x - 1 = 0 or x + 4 = 0. Solving for x gives us x = 1 and x = -4. These are the x-intercepts of the function. The vertex of the parabola is also important. Since the leading coefficient of (x - 1)(x + 4) is positive, the parabola opens upwards. This means that the vertex is the lowest point on the graph. The combination of the x-intercepts at x = 1 and x = -4, and the fact that the parabola opens upwards, allows us to pinpoint Graph D as the correct representation of f(x) = (x - 1)(x + 4).

The statement that is true about the function f(x) = (x - 4)(x + 1) is the one that accurately describes the function's behavior and key characteristics. The x-intercepts of the function are (4, 0) and (-1, 0). The vertex of the parabola is the lowest point on the graph. Since the leading coefficient of (x - 4)(x + 1) is positive, the parabola opens upwards. So, the correct statement is "The function has a minimum value at the vertex and two x-intercepts."

The graph of a quadratic function is a parabola. The x-intercepts of the function are the points where the graph crosses the x-axis. To find the x-intercepts, we set the function equal to zero and solve for x. In this case, the function is (x + 2)(x + 6) = 0. To make this equation true, one or both of the factors must be zero. Therefore, x + 2 = 0 or x + 6 = 0. Solving for x gives us x = -2 and x = -6. These are the x-intercepts of the function. The vertex of the parabola is also important. Since the leading coefficient of (x + 2)(x + 6) is positive, the parabola opens upwards. This means that the vertex is the lowest point on the graph. The combination of the x-intercepts at x = -2 and x = -6, and the fact that the parabola opens upwards, allows us to pinpoint the correct statement: "The function has a minimum value at the vertex and two x-intercepts."