Question

find the y - intercept, the axis of symmetry, and the vertex of the graph of the function. \\( f(x) = 3x^2 + 18x + 4 \\) \\( \text{the y - intercept is } (0, 4) \\). (type an ordered pair.) \\( \text{the axis of symmetry is } x = 0 \\). (simplify your answer. type an equation.)

Explanation:

Step1: Find the y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(f(x)=3x^{2}+18x + 4\).
Substitute \(x = 0\) into the function: \(f(0)=3(0)^{2}+18(0)+4=4\). So the y - intercept is \((0,4)\).

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 \(f(x)=3x^{2}+18x + 4\), we have \(a = 3\) and \(b = 18\).
Substitute \(a = 3\) and \(b = 18\) into the formula for the axis of symmetry: \(x=-\frac{18}{2\times3}=-\frac{18}{6}=- 3\).

Step3: Find the vertex

The x - coordinate of the vertex is the same as the axis of symmetry, so \(x=-3\).
To find the y - coordinate of the vertex, we substitute \(x = - 3\) into the function \(f(x)\):
\(f(-3)=3(-3)^{2}+18(-3)+4=3\times9-54 + 4=27-54 + 4=-23\).
So the vertex is \((-3,-23)\).

Answer:

• y - intercept: \((0,4)\)
• Axis of symmetry: \(x=-3\)
• Vertex: \((-3,-23)\)