Question

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

y - intercept: $y = 19$
vertex: $(9, 10)$
x - intercepts: $x = -9$ and $x = -10$
x - intercepts: $x = 9$ and $x = 10$

Explanation:

Step1: Recall x - intercept definition

To find x - intercepts, set \( f(x)=0 \). Given \( f(x)=(x - 9)(x - 10) \), we set \( (x - 9)(x - 10)=0 \).

Step2: Solve for x

Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So, \( x-9 = 0 \) gives \( x = 9 \), and \( x - 10=0 \) gives \( x = 10 \).

Step3: Analyze other options

• For y - intercept: Set \( x = 0 \), \( f(0)=(0 - 9)(0 - 10)=(-9)\times(-10)=90

eq19 \), so y - intercept is not 19.

• For vertex: The function \( f(x)=(x - 9)(x - 10)=x^{2}-19x + 90 \). The x - coordinate of the vertex of a quadratic \( ax^{2}+bx + c \) is \( x=-\frac{b}{2a} \). Here \( a = 1 \), \( b=-19 \), so \( x=\frac{19}{2}=9.5 \), and \( f(9.5)=(9.5 - 9)(9.5 - 10)=(0.5)\times(-0.5)=-0.25 \), so vertex is not \( (9,10) \).
• The orange option has incorrect x - intercepts as we found \( x = 9 \) and \( x = 10 \), not \( x=-9 \) and \( x=-10 \).

Answer:

x - intercepts: \( x = 9 \) and \( x = 10 \) (the teal option)