Question

which equation choice could represent the graph shown below?
answer
$f(x) = x(x - 5)(x - 5)$ $f(x) = (x + 5)(x + 5)$
$f(x) = (x - 5)(x - 5)$ $f(x) = (x + 5)(x - 5)$

Explanation:

Step1: Analyze the x-intercept

The graph touches the x - axis at \(x = 5\) (it has a repeated root at \(x=5\)). For a polynomial function, if the graph touches the x - axis at a point \(x = a\), then \((x - a)\) is a factor with an even multiplicity.

Step2: Analyze each option

• Option 1: \(f(x)=x(x - 5)(x - 5)\) has roots at \(x = 0\) (multiplicity 1) and \(x=5\) (multiplicity 2). But the graph does not cross or touch the x - axis at \(x = 0\), so this is incorrect.
• Option 2: \(f(x)=(x + 5)(x + 5)\) has a root at \(x=-5\), but the graph is not near \(x = - 5\), so this is incorrect.
• Option 3: \(f(x)=(x - 5)(x - 5)=(x - 5)^2\). This is a quadratic function (a parabola) with a repeated root at \(x = 5\), which matches the graph (the graph touches the x - axis at \(x = 5\) and is a parabola opening upwards).
• Option 4: \(f(x)=(x + 5)(x - 5)=x^2-25\) has roots at \(x=-5\) and \(x = 5\), and the graph of this function is a parabola opening upwards with x - intercepts at \(x=-5\) and \(x = 5\), but our graph only touches at \(x = 5\) and has no intercept at \(x=-5\), so this is incorrect.

Answer:

\(f(x)=(x - 5)(x - 5)\) (or \(f(x)=(x - 5)^2\))