Question

find the y - intercept, the axis of symmetry, and the vertex of the graph of the function.
$f(x)=3x^{2}+18x + 5$
the y - intercept is (0,5). (type an ordered pair.)
the axis of symmetry is $x=-3$. (simplify your answer. type an equation.)
the vertex is $(-3,18)$. (type an ordered pair.)

Explanation:

Step1: Find the y - intercept

To find the y - intercept of a function \(f(x)\), we set \(x = 0\) and solve for \(f(0)\).
For the function \(f(x)=3x^{2}+18x + 5\), when \(x = 0\):
\(f(0)=3(0)^{2}+18(0)+5=5\)
So the y - intercept is the ordered pair \((0,5)\).

Step2: Find the axis of symmetry

For a quadratic function in the form \(f(x)=ax^{2}+bx + c\) (\(a
eq0\)), the equation of the axis of symmetry is given by \(x=-\frac{b}{2a}\).
In the function \(f(x)=3x^{2}+18x + 5\), \(a = 3\) and \(b = 18\).
Substitute \(a = 3\) and \(b = 18\) into the formula \(x=-\frac{b}{2a}\):
\(x=-\frac{18}{2\times3}=-\frac{18}{6}=- 3\)
So the axis of symmetry is the line \(x=-3\).

Step3: Find the vertex

The x - coordinate of the vertex of a quadratic function \(f(x)=ax^{2}+bx + c\) is the same as the axis of symmetry, i.e., \(x =-\frac{b}{2a}\). We already found that \(x=-3\).
To find the y - coordinate of the vertex, we substitute \(x=-3\) into the function \(f(x)\):
\(f(-3)=3(-3)^{2}+18(-3)+5=3\times9-54 + 5=27-54 + 5=-22\)? Wait, there is a mistake in the original given vertex. Let's recalculate:
\(f(-3)=3\times(-3)^{2}+18\times(-3)+5=3\times9-54 + 5=27-54 + 5=-22\). But according to the problem's given (maybe a typo, but we follow the calculation steps). Wait, maybe the original function is different? Wait, no, let's check again.
Wait, \(3x^{2}+18x + 5\), when \(x=-3\):
\(3\times(-3)^{2}=3\times9 = 27\), \(18\times(-3)=-54\), so \(27-54 + 5=-22\). But the problem says the vertex is \((-3,18)\), which is incorrect. But we will follow the correct method.
The x - coordinate of the vertex is \(x=-3\) (from the axis of symmetry). Then we find \(f(-3)\):
\(f(-3)=3(-3)^{2}+18(-3)+5=27-54 + 5=-22\). So the vertex should be \((-3,-22)\). But if we go with the problem's given (assuming maybe a typo in the function, but we proceed with the correct formula).

Answer:

• The y - intercept is \((0,5)\).
• The axis of symmetry is \(x = - 3\).
• The vertex is \((-3,-22)\) (Note: There is a discrepancy with the given vertex in the problem, the correct vertex using the function \(f(x)=3x^{2}+18x + 5\) is \((-3,-22)\))