The function $f$ is defined by $f(x)=a x^{2}+b x+c$ , where $a, b$ , and $c$ are constants and $1<a<4$ . The graph of $y=f(x)$ in the $x y$ -plane passes through points. $(11,0)$ and $(-2,0)$ . If $a$ is an integer, what could be the value of $a+b$ ?

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Algebra 2

Find the vertex of the transformed function

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Q. The function

f

is defined by

f(x)=a x^{2}+b x+c

, where

a, b

, and

c

are constants and

1<a<4

. The graph of

y=f(x)

in the

x y

-plane passes through points.

(11,0)

and

(-2,0)

. If

a

is an integer, what could be the value of

a+b

Set up equations: Since the graph of $y=f(x)$ passes through the points $(11,0)$ and $(-2,0)$ , we can set up two equations using these points to find the relationship between $a$ , $b$ , and $c$ . $f(11) = a(11)^2 + b(11) + c = 0$ $f(-2) = a(-2)^2 + b(-2) + c = 0$
Substitute x-values: Let's substitute the x-values from the points into the equations to get: $\newline$ $a(11)^2 + b(11) + c = 0$ $\newline$ $a(-2)^2 + b(-2) + c = 0$
Simplify equations: Now we simplify the equations: $\newline$ $121a + 11b + c = 0$ $\newline$ $4a - 2b + c = 0$
Express $c$ in terms: We have two equations with three unknowns, which means we cannot find the exact values of $a$ , $b$ , and $c$ without additional information. However, we can express $c$ in terms of $a$ and $b$ using one of the equations. $\newline$ Let's use the second equation to express $c$ : $\newline$ $c = -4a + 2b$
Substitute $c$ into first equation: Substitute $c$ from the fourth step into the first equation: $\newline$ $121a + 11b - 4a + 2b = 0$ $\newline$ $117a + 13b = 0$
Find possible values of $a$ : We need to find the values of $a$ and $b$ such that $a$ is an integer and $1 < a < 4$ . Since $a$ must be an integer, let's list the possible values of $a$ : $2$ , $3$ .
Try $a = 2$ : First, let's try $a = 2$ and solve for $b$ using the equation $117a + 13b = 0$ :
$117(2) + 13b = 0$
$234 + 13b = 0$
$13b = -234$
$b = -234 / 13$
$b = -18$
Try $a = 3$ : Now let's check if $a = 3$ gives us an integer value for $b$ :
$117(3) + 13b = 0$
$351 + 13b = 0$
$13b = -351$
$b = -351 / 13$
$b = -27$
Calculate $a + b$ : Both $a = 2$ and $a = 3$ give us integer values for $b$ . Now we can calculate $a + b$ for each case: $\newline$ For $a = 2$ , $b = -18$ , so $a + b = 2 - 18 = -16$ . $\newline$ For $a = 3$ , $b = -27$ , so $a = 2$ $0$ .
Check valid values: Since the question asks for possible values of $a + b$ , we have two possible answers: $-16$ and $-24$ . However, we need to ensure that $a + b$ is a single value, so we must check if both values of $a$ are valid given the constraints of the problem.
Check valid values: Since the question asks for possible values of $a + b$ , we have two possible answers: $-16$ and $-24$ . However, we need to ensure that $a + b$ is a single value, so we must check if both values of $a$ are valid given the constraints of the problem.The constraints are $1 < a < 4$ and $a$ is an integer. Both $a = 2$ and $a = 3$ satisfy these constraints. Therefore, both $-16$ and $-24$ are valid answers for $a + b$ .