Question

Which of these statements accurately describes the graph of the function f(x)=x2+14x+49f(x)=x2+14x+49 ? A The graph passes through the x-axis at -7. B The graph touches the x-axis at -7. C The graph passes through the x-axis at 7. D The graph touches the x-axis at 7.

Answer 1

A point belongs to the x axis if its y coordinate equals zero.

The points on a graph are in the form $(x,f(x))$, so these points are on the x axis if and only if $f(x)=0$

In this case, we have

$f(x) = 0 \iff x^2+14x+49=0$

You can observe that your expression is actually a squared binomial: using

$(a+b)^2 = a^2+2ab+b^2$

you can notice that

$(x+7)^2 = x^2+14x+49$

So, you have

$x^2+14x+49=0 \iff (x-7)^2 = 0 \iff x=7$

Now, how we decide if this function "touches" or "passes through" the x-axis at x=7? Well, since our function is a square, it is never negative. So, this graph can't cross the x-axis, but rater touch it from above. The parabola has a U shape, and the point of minimum lies on the x axis.

So, the graph touches the x axis at x=7.