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Answer:
• Vertex Form: f(x)=(x+5)²-7
,• Vertex: (-5, -7)
,• Axis of symmetry: x=-5
,• Graph: See below
Explanation:
Given the function:
[tex]f(x)=x^2+10x+18[/tex]Wo want t wrerite the function in the vertex form y=a(x-h)².
In order to do this, complete the square for x.
[tex]\begin{gathered} f(x)=x^2+10x+18 \\ \text{Divide the coefficient of x by 2, square it and add it to both sides.} \\ f(x)+5^2=x^2+10x+5^2+18 \\ f(x)+25=(x+5)^2+18 \\ \text{ Subtract 25 from both sides} \\ f(x)+25-25=(x+5)^2+18-25 \\ f(x)=(x+5)^2-7 \end{gathered}[/tex]The vertex form of the function is:
[tex]f(x)=(x+5)^2-7[/tex]The vertex of the function:
[tex]\left(h,k\right)=(-5,-7)[/tex]The axis of symmetry is the x-value at the vertex, therefore, the axis of symmetry is:
[tex]x=-5[/tex]Graph
In odert to graph th function, find two otherpoints ton the graph.
[tex]\begin{gathered} \text{ When }x=-8,f(-8)=(-8)^2+10(-8)+18=2\implies(-8,2) \\ \text{When }x=-2,f(-2)=(-2)^2+10(-2)+18=2\implies(-2,2) \\ \text{When }x=0,f(0)=(0)^2+10(0)+18=18\implies(0,18) \end{gathered}[/tex]Plot the vertex, (-5,-7) and the poins (-8,2), (-2,2) and (0,18).
The graph of f(x) is attached below:
