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The equation
y=x^(2)+6x+b is graphed in the
xy-plane. If the vertex of the graph of the equation is at
(-3,0), what is the value of
b ?

The equation $y=x^{2}+6 x+b$ is graphed in the $x y$ -plane. If the vertex of the graph of the equation is at $(-3,0)$ , what is the value of $b$ ?

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Write a quadratic function in vertex form

Full solution

Q. The equation

y=x^{2}+6 x+b

is graphed in the

x y

-plane. If the vertex of the graph of the equation is at

(-3,0)

, what is the value of

b

?

Identify vertex form: Identify the vertex form of a parabola. $\newline$ The vertex form of a parabola is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
Write equation with given vertex: Use the given vertex $(-3, 0)$ to write the equation in vertex form. $\newline$ Since the vertex is $(-3, 0)$ , we have $h = -3$ and $k = 0$ . Thus, the vertex form of the equation is $y = a(x - (-3))^2 + 0$ , which simplifies to $y = a(x + 3)^2$ .
Expand vertex form: Expand the vertex form to compare it with the given equation. $\newline$ Expanding $y = a(x + 3)^2$ gives us $y = a(x^2 + 6x + 9)$ . Since the given equation is $y = x^2 + 6x + b$ , we can compare the coefficients to find the value of $a$ and $b$ .
Compare coefficients for $a$ : Compare the coefficients of $x^2$ and $x$ to find the value of $a$ . $\newline$ The coefficient of $x^2$ in both the expanded vertex form and the given equation is $1$ , so $a = 1$ . The coefficient of $x$ is also the same ( $6$ ), which confirms that $a = 1$ is correct.
Determine value of $b$ : Determine the value of $b$ by comparing the constant terms. $\newline$ In the expanded vertex form, the constant term is $9a$ , which is $9$ since $a = 1$ . For the vertex form to match the given equation, the constant term must also be $b$ . Therefore, $b = 9$ .
Verify vertex with $b = 9$ : Verify that the vertex $(-3, 0)$ is correct with $b = 9$ . $\newline$ Substitute $x = -3$ into the given equation $y = x^2 + 6x + 9$ to check if $y$ equals $0$ . $y = (-3)^2 + 6(-3) + 9 = 9 - 18 + 9 = 0$ . This confirms that the vertex is indeed $(-3, 0)$ when $b = 9$ .

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