Question

heather is writing a quadratic function that represents a parabola that touches but does not cross the x-axis at x = -6. which function could heather be writing? \\( f(x) = x^2 + 36x + 12 \\) \\( f(x) = x^2 - 36x - 12 \\) \\( f(x) = -x^2 + 12x + 36 \\) \\( f(x) = -x^2 - 12x - 36 \\)

Explanation:

Step1: Recall the property of a parabola touching the x - axis

A parabola touches but does not cross the x - axis at a point when the quadratic function has a repeated root at that point. For a quadratic function \(y = ax^{2}+bx + c\), the discriminant \(\Delta=b^{2}-4ac = 0\) when there is a repeated root, and the root is \(x=-\frac{b}{2a}\). Here, the repeated root is \(x = - 6\), so \(-\frac{b}{2a}=-6\), which implies \(b = 12a\) (if \(a
eq0\)). Also, the discriminant \(b^{2}-4ac=0\).

Step2: Analyze each option

• Option 1: \(f(x)=x^{2}+36x + 12\)

Here, \(a = 1\), \(b = 36\), \(c = 12\).
First, check \(-\frac{b}{2a}=-\frac{36}{2\times1}=-18
eq - 6\). So this is not the function.

• Option 2: \(f(x)=x^{2}-36x - 12\)

Here, \(a = 1\), \(b=-36\), \(c = - 12\).
\(-\frac{b}{2a}=-\frac{-36}{2\times1}=18
eq - 6\). So this is not the function.

• Option 3: \(f(x)=-x^{2}+12x + 36\)

Here, \(a=-1\), \(b = 12\), \(c = 36\).
\(-\frac{b}{2a}=-\frac{12}{2\times(-1)} = 6
eq - 6\). So this is not the function.

• Option 4: \(f(x)=-x^{2}-12x - 36\)

Here, \(a=-1\), \(b=-12\), \(c=-36\).
First, check \(-\frac{b}{2a}=-\frac{-12}{2\times(-1)}=-6\).
Now check the discriminant: \(b^{2}-4ac=(-12)^{2}-4\times(-1)\times(-36)=144 - 144=0\).
So the discriminant is 0 and the vertex (the point where it touches the x - axis) is at \(x=-6\).

Answer:

\(f(x)=-x^{2}-12x - 36\) (the fourth option)