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The functions
f(x)=-2.5(x+2)^(2)+8 and
g(x)=-2.5(x+2)^(2)+b are graphed in the
xy-plane as
y=f(x) and
y=g(x). Let
a be the
y-coordinate of the vertex of function
f and
b be the
y-coordinate of the vertex of function
g. If
a is 5 less than
b, then what is the value of
b ?

The functions $f(x)=-2.5(x+2)^{2}+8$ and $g(x)=-2.5(x+2)^{2}+b$ are graphed in the $x y$ -plane as $y=f(x)$ and $y=g(x)$ . Let $a$ be the $y$ -coordinate of the vertex of function $f$ and $b$ be the $y$ -coordinate of the vertex of function $g(x)=-2.5(x+2)^{2}+b$ $0$ . If $a$ is $5$ less than $b$ , then what is the value of $b$ ?

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Write and solve direct variation equations

Full solution

Q. The functions

f(x)=-2.5(x+2)^{2}+8

and

g(x)=-2.5(x+2)^{2}+b

are graphed in the

x y

-plane as

y=f(x)

and

y=g(x)

. Let

a

be the

y

-coordinate of the vertex of function

f

and

b

be the

y

-coordinate of the vertex of function

g(x)=-2.5(x+2)^{2}+b

0

. If

a

is

5

less than

b

, then what is the value of

b

?

Vertex form of quadratic function: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. For the function $f(x) = -2.5(x + 2)^2 + 8$ , the vertex $(h, k)$ is $(-2, 8)$ .
Finding the y-coordinate of the vertex: The y-coordinate of the vertex of function $f$ is $8$ , which is represented by $a$ in the problem statement.
Relationship between $a$ and $b$ : According to the problem, $a$ is $5$ less than $b$ . This can be written as $a = b - 5$ .
Substituting $a$ to find $b$ : Substitute the value of $a$ (which is $8$ ) into the equation $a = b - 5$ to find $b$ . $\newline$ $8 = b - 5$
Solving for b: Add $5$ to both sides of the equation to solve for $b$ .
$8 + 5 = b$
$13 = b$

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