For Question 2: Use the graph and equation provided to find the requested information You can earn: - 2 pts for each blank (1 pt will be given for an incomplete or poorly formatted response) 2. Given f(x)=(1)/(2) x^2+2 x-6 Identify the following information: y -intercept & x -intercept(s) & vertex & Axis of symmetry &

To find the requested information for the quadratic function \( f(x) = \frac{1}{2}x^2 + 2x - 6 \), we will analyze the equation step by step.<br /><br />### \( y \)-intercept:<br />The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function:<br /><br />\[ f(0) = \frac{1}{2}(0)^2 + 2(0) - 6 = -6 \]<br /><br />So, the \( y \)-intercept is \(-6\).<br /><br />### \( x \)-intercept(s):<br />The \( x \)-intercepts occur where \( f(x) = 0 \). Set the equation to zero and solve for \( x \):<br /><br />\[ \frac{1}{2}x^2 + 2x - 6 = 0 \]<br /><br />Multiply through by 2 to clear the fraction:<br /><br />\[ x^2 + 4x - 12 = 0 \]<br /><br />Now, factor the quadratic:<br /><br />\[ (x + 6)(x - 2) = 0 \]<br /><br />Setting each factor equal to zero gives:<br /><br />\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]<br />\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]<br /><br />Thus, the \( x \)-intercepts are \(-6\) and \(2\).<br /><br />### Vertex:<br />The vertex of a parabola in the form \( ax^2 + bx + c \) can be found using the formula for the \( x \)-coordinate of the vertex: <br /><br />\[ x = -\frac{b}{2a} \]<br /><br />Here, \( a = \frac{1}{2} \) and \( b = 2 \). So,<br /><br />\[ x = -\frac{2}{2 \times \frac{1}{2}} = -2 \]<br /><br />Substitute \( x = -2 \) back into the function to find the \( y \)-coordinate:<br /><br />\[ f(-2) = \frac{1}{2}(-2)^2 + 2(-2) - 6 \]<br />\[ = \frac{1}{2}(4) - 4 - 6 \]<br />\[ = 2 - 4 - 6 \]<br />\[ = -8 \]<br /><br />Therefore, the vertex is \((-2, -8)\).<br /><br />### Axis of symmetry:<br />The axis of symmetry for a parabola given by \( ax^2 + bx + c \) is the vertical line that passes through the vertex. It has the equation:<br /><br />\[ x = -\frac{b}{2a} \]<br /><br />From our previous calculation, this is:<br /><br />\[ x = -2 \]<br /><br />So, the axis of symmetry is \( x = -2 \).<br /><br />In summary, the information is as follows:<br /><br />\[<br />\begin{array}{|l|l|}<br\ />\hline<br\ />\ y\ -intercept\ &\ -6\ \\<br\ />\hline<br\ />\ x\ -intercept(s)\ &\ -6,\ 2\ \\<br\ />\hline<br\ />vertex\ &\ (-2,\ -8)\ \\<br\ />\hline<br\ />Axis\ of\ symmetry\ &\ x\ =\ -2\ \\<br\ />\hline<br\ />\end{array}<br />\]