Question

what information can i find from this form of the equation?
$f(x) = (x - 9)(x - 10)$

1. x-intercepts: $x = -9$ and $x = -10$
2. x-intercepts: $x = 9$ and $x = 10$
3. y-intercept: $y = 19$
4. vertex: $(9, 10)$

Explanation:

Step1: Recall x - intercept definition

To find x - intercepts, set \(y = f(x)=0\). The function is \(f(x)=(x - 9)(x - 10)\).
Set \((x - 9)(x - 10)=0\).

Step2: Solve for x

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
For \((x - 9)(x - 10)=0\), we have \(x-9 = 0\) or \(x - 10=0\).
Solving \(x-9 = 0\) gives \(x = 9\), and solving \(x - 10=0\) gives \(x = 10\).
So the x - intercepts are \(x = 9\) and \(x = 10\).

For the y - intercept:

Step1: Recall y - intercept definition

To find the y - intercept, set \(x = 0\) in the function \(f(x)=(x - 9)(x - 10)\).

Step2: Calculate \(f(0)\)

\(f(0)=(0 - 9)(0 - 10)=(-9)\times(-10)=90
eq19\).
For the vertex:
The function \(f(x)=(x - 9)(x - 10)=x^{2}-19x + 90\). The x - coordinate of the vertex of a quadratic function \(ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\). Here \(a = 1\), \(b=-19\), so \(x=\frac{19}{2}=9.5\). Then \(f(9.5)=(9.5 - 9)(9.5 - 10)=(0.5)\times(-0.5)=- 0.25\). So the vertex is \((9.5,-0.25)
eq(9,10)\).

Answer:

The correct option is the one with x - intercepts: \(x = 9\) and \(x = 10\) (the blue - colored option with this description).