2. f(x)=x^2-4 x-5 Identify the following information: Vertex & Axis of Symmetry & Increasing Interval & Decreasing Interval & End Behavior &

To analyze the quadratic function \( f(x) = x^2 - 4x - 5 \), we can identify the following characteristics:<br /><br />1. **Vertex**: <br /> The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -4 \), and \( c = -5 \).<br /><br /> \[<br /> x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2<br /> \]<br /><br /> Substitute \( x = 2 \) back into the function to find the y-coordinate:<br /><br /> \[<br /> f(2) = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9<br /> \]<br /><br /> Therefore, the vertex is at \( (2, -9) \).<br /><br />2. **Axis of Symmetry**: <br /> The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Thus, it is given by the equation \( x = 2 \).<br /><br />3. **Increasing Interval**: <br /> A quadratic function \( f(x) = ax^2 + bx + c \) with \( a > 0 \) decreases to the left of the vertex and increases to the right. Since the vertex is at \( x = 2 \), the function is increasing on the interval \( (2, \infty) \).<br /><br />4. **Decreasing Interval**: <br /> Similarly, the function is decreasing on the interval \( (-\infty, 2) \).<br /><br />5. **End Behavior**: <br /> For a quadratic function \( f(x) = ax^2 + bx + c \), as \( x \to \pm\infty \), the behavior depends on the sign of \( a \). Here, \( a = 1 > 0 \), so:<br /><br /> - As \( x \to -\infty \), \( f(x) \to \infty \).<br /> - As \( x \to \infty \), \( f(x) \to \infty \).<br /><br />In summary:<br /><br />\[<br />\begin{array}{|l|l|}<br\ />\hline\ <br\ />Vertex\ &\ (2,\ -9)\ \\<br\ />\hline\ <br\ />Axis\ of\ Symmetry\ &\ x\ =\ 2\ \\<br\ />\hline\ <br\ />Increasing\ Interval\ &\ (2,\ \infty)\ \\<br\ />\hline\ <br\ />Decreasing\ Interval\ &\ (-\infty,\ 2)\ \\<br\ />\hline\ <br\ />End\ Behavior\ &\ \text{As\ }\ x\ \to\ -\infty,\ f(x)\ \to\ \infty;\ \text{\ as\ }\ x\ \to\ \infty,\ f(x)\ \to\ \infty\ \\<br\ />\hline<br\ />\end{array}<br />\]