Question

find the y-intercept, the axis of symmetry, and the vertex of the graph of the function.\\( f(x) = -x^2 - 2x - 9 \\)\\( \\)\\( \\)the y-intercept is \\( (0, -9) \\). (type an ordered pair.)\\( \\)the axis of symmetry is \\( x = -1 \\). (simplify your answer. type an equation.)\\( \\)the vertex is \\( (-1, -2) \\). (type an ordered pair.)

Explanation:

Step1: Find y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(f(x)=-x^{2}-2x - 9\).
Substitute \(x = 0\) into the function: \(f(0)=-(0)^{2}-2(0)-9=-9\). So the y - intercept is the ordered pair \((0,-9)\).

Step2: Find 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 \(x =-\frac{b}{2a}\).
In the function \(f(x)=-x^{2}-2x - 9\), \(a=-1\) and \(b = - 2\).
Substitute \(a=-1\) and \(b=-2\) into the formula: \(x=-\frac{-2}{2\times(-1)}=-\frac{2}{2}=-1\). So the axis of symmetry is \(x=-1\).

Step3: Find the vertex

The x - coordinate of the vertex is the same as the axis of symmetry, so \(x=-1\).
To find the y - coordinate of the vertex, we substitute \(x = - 1\) into the function \(f(x)=-x^{2}-2x - 9\).
\(f(-1)=-(-1)^{2}-2(-1)-9=-1 + 2-9=-8\)? Wait, there is a mistake in the original answer. Let's recalculate:
\(f(-1)=-(-1)^{2}-2\times(-1)-9=-1 + 2-9=-8\). Wait, the original answer has \((-1,-2)\) which is wrong. Let's do it correctly.
Wait, \(f(x)=-x^{2}-2x - 9\), we can also complete the square.
\(f(x)=-(x^{2}+2x)-9=-(x^{2}+2x + 1-1)-9=-(x + 1)^{2}+1-9=-(x + 1)^{2}-8\).
So when \(x=-1\), \(f(-1)=-8\). So the vertex is \((-1,-8)\).

Wait, let's check the original calculation again. If we use the formula for the vertex form \(y=a(x - h)^{2}+k\), where \((h,k)\) is the vertex.
From \(f(x)=-x^{2}-2x - 9\), \(a=-1\), \(b=-2\), \(c = - 9\).
The x - coordinate of the vertex \(h=-\frac{b}{2a}=-\frac{-2}{2\times(-1)}=-1\).
Then \(k=f(h)=f(-1)=-(-1)^{2}-2\times(-1)-9=-1 + 2-9=-8\). So the vertex is \((-1,-8)\). The original answer has an error.

But according to the problem - solving process:

Step1: Y - intercept

Set \(x = 0\), \(f(0)=-0-0 - 9=-9\), so \((0,-9)\) is correct.

Step2: Axis of symmetry

\(x=-\frac{b}{2a}=-\frac{-2}{2\times(-1)}=-1\), correct.

Step3: Vertex

x - coordinate is \(-1\), substitute into \(f(x)\): \(f(-1)=-(-1)^{2}-2\times(-1)-9=-1 + 2-9=-8\). So the vertex is \((-1,-8)\).

Answer:

• Y - intercept: \((0,-9)\)
• Axis of symmetry: \(x = - 1\)
• Vertex: \((-1,-8)\) (Note: The original vertex answer \((-1,-2)\) is incorrect. The correct y - coordinate when \(x=-1\) is \(f(-1)=-(-1)^2-2\times(-1)-9=-1 + 2-9=-8\))