Problem

Compare the zeros, \(x\text{-}\text{intercepts}\), and \(y\text{-}\text{intercepts}\) of \(f(x)~=~(x~-~2)(x~+~3)\) with graph \(g\), whose \(x\text{-}\text{intercepts}\) are \(-3\) and \(2\) and \(y\text{-}\text{intercept}\) is \(6\). State whether the intercept data establish that the functions are identical.

Two distinct quadratic curves crossing the x-axis at negative 3 and 2.

Big Picture

What this problem is really about

Shared zeros tell you that two graphs cross the x-axis at the same places, but they do not determine the rest of either function. Compare another easy feature, such as the y-intercept, to test whether the curves really agree. A single input with different outputs is enough to disprove equality.

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Both have zeros 3 and −2, so their x-intercepts match. f has y-intercept −6 while g has y-intercept 6, so they are not the same function.

Both have zeros −3 and 2, so their x-intercepts match. f has y-intercept −6 while g has y-intercept 6, so they are not the same function.

Both have zeros −3 and 2, so their x-intercepts match. f has y-intercept −6 while g has y-intercept 6, so they are the same function.

Both have zeros −3 and 2, so their x-intercepts match. f has y-intercept 6 while g has y-intercept 6, so they are not the same function.

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