Let f(x) be a quadratic function and c constant. Which of the following statements is correct? (Statements in body of question)

Question

a) There is a unique value of c such that y = f(x) - c has a double root.

b) There is unique value of c such that y = f(x-c) has a double root.

However, I believe that both of the statements are correct. I considered when f(x) = x^2 and c = 0, and both times I found that the function touched the x-axis at 0 and nowhere else, which suggested to me that there was a double root at 0.

I know the previous statement is incorrect, but I don't know why.

Answer 1

$(a)$ is correct but $(b)$ is incorrect. If $f(x)$ has no real roots, then $f(x-c)$ has no real roots either for any $c$. For part $(a)$, you can just subtract the value of $f$ at the vertex.

Answer 2

$y=f(x)-c$ moves the parabola up or down. There is only one value of $c$ that will put the vertex of the parabola coincident with the $x$-axis.

$y=f(x-c)$ moves the parabola left or right. No matter what the value of $c$ is, the number of roots will not change.