Determine whether the statement is true or false. Justify your answer. The function f(x)=a(x-5)^

step 1 Hello everyone in this question we have to tell whether given statement is true or false. So, qu

Determine whether the statement is true or false. Justify...

Key Concepts

x-intercepts of a Function

An x-intercept of a function is the point where its graph crosses the x-axis, meaning the y-value (or f(x)) is zero at that point. To find the x-intercepts for any function, one sets f(x) equal to zero and solves for x. In the context of quadratic functions, this often involves solving a quadratic equation, which can yield two, one, or no real solutions depending on the discriminant.

Unique x-intercept in Parabolas

For a quadratic function, a unique x-intercept occurs when the parabola just touches the x-axis rather than cutting through it. This special case happens when the vertex lies exactly on the x-axis (i.e., when k = 0 in the vertex form) and the vertex is the only solution to f(x) = 0. This results in a repeated root, indicating a single intersection point with the x-axis.

Quadratic Function in Vertex Form

A quadratic function expressed in vertex form, such as f(x) = a(x - h)^2 + k, clearly shows the vertex of the parabola at the point (h, k). This form simplifies the analysis of the graph’s features, including the axis of symmetry, the direction of opening, and the location of any intercepts. Knowing the vertex location is crucial to determining the behavior of the function around that point.

Impact of a Nonzero Parameter a

In the quadratic function f(x) = a(x - 5)^2, the parameter a determines the direction (upward for a > 0 or downward for a < 0) and the width of the parabola. Despite how a scales the graph, as long as a is nonzero, the fact that the vertex is set at (5, 0) guarantees that the parabola touches the x-axis only at that point, leading to exactly one x-intercept.

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