Question

consider the function $g(x)=x^2 + 8x + 18$
graph $g(x)$
what is the vertex of $g$?
what is the equation of the line of symmetry of $g$?
$g$ has a select an answer of
the $x$-intercept(s) of $g$ is/are
the $y$-intercept of $g$ is
for this question give all rational answers as fractions or integers and all irrational answers rounded to 2 decimal places.
if there are no intercepts, you can enter dne in that box to say that the intercepts \do not exist.\
question help: video written example
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Explanation:

1. Vertex of \( g(x) \)

Step1: Complete the square for \( g(x)=x^2 + 8x + 18 \)

To find the vertex, we complete the square. For a quadratic function \( ax^2+bx+c \), we use the formula \( x^2+bx=(x+\frac{b}{2})^2-\frac{b^2}{4} \). Here, \( a = 1 \), \( b = 8 \), so:
\( g(x)=x^2 + 8x + 18=(x^2 + 8x + 16)-16 + 18 \)

Step2: Simplify the expression

Simplify the above expression:
\( g(x)=(x + 4)^2+2 \)
The vertex form of a quadratic function is \( y=a(x - h)^2+k \), where the vertex is \( (h,k) \). Comparing \( (x + 4)^2+2 \) with \( a(x - h)^2+k \), we have \( h=-4 \), \( k = 2 \). So the vertex is \( (-4,2) \).

2. Line of Symmetry of \( g(x) \)

Step1: Recall the formula for line of symmetry

For a quadratic function in the form \( y=a(x - h)^2+k \), the line of symmetry is \( x = h \). From the vertex form we found earlier, \( g(x)=(x + 4)^2+2 \), so \( h=-4 \).

Step2: Determine the line of symmetry

The line of symmetry is \( x=-4 \).

3. Minimum/Maximum Value of \( g(x) \)

Step1: Analyze the coefficient of \( x^2 \)

For the quadratic function \( g(x)=x^2 + 8x + 18 \), the coefficient of \( x^2 \) is \( a = 1>0 \), so the parabola opens upwards.

Step2: Determine the type of extremum and its value

When the parabola opens upwards, the function has a minimum value at the vertex. From the vertex \( (-4,2) \), the minimum value is \( 2 \).

4. \( x \)-intercept(s) of \( g(x) \)

Answer:

s:

• Vertex of \( g \): \((-4, 2)\)
• Equation of the line of symmetry of \( g \): \( x=-4 \)
• \( g \) has a minimum of \( 2 \)
• The \( x \)-intercept(s) of \( g \): DNE
• The \( y \)-intercept of \( g \): \( 18 \)