. Given the function f(x)=5x^2+10x-15 convert it to vertex and factored form, and list the key properties of the function, and then graph it (graph paper on next page) [16]

## Step 1: Convert to Vertex Form<br />### To convert the quadratic function $f(x) = 5x^2 + 10x - 15$ to vertex form, we complete the square. The vertex form of a quadratic function is given by $a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.<br /><br />Given:<br />\[ f(x) = 5x^2 + 10x - 15 \]<br /><br />First, factor out the coefficient of $x^2$ from the first two terms:<br />\[ f(x) = 5(x^2 + 2x) - 15 \]<br /><br />Next, complete the square inside the parentheses:<br />\[ x^2 + 2x \]<br />Add and subtract $(\frac{2}{2})^2 = 1$ inside the parentheses:<br />\[ x^2 + 2x + 1 - 1 = (x+1)^2 - 1 \]<br /><br />Now substitute back into the function:<br />\[ f(x) = 5((x+1)^2 - 1) - 15 \]<br />\[ f(x) = 5(x+1)^2 - 5 - 15 \]<br />\[ f(x) = 5(x+1)^2 - 20 \]<br /><br />Thus, the vertex form is:<br />\[ f(x) = 5(x - (-1))^2 - 20 \]<br /><br />## Step 2: Convert to Factored Form<br />### To convert the quadratic function to factored form, we find the roots of the equation by solving $5x^2 + 10x - 15 = 0$.<br /><br />Using the quadratic formula:<br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br />where $a = 5$, $b = 10$, and $c = -15$:<br /><br />\[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 5 \cdot (-15)}}{2 \cdot 5} \]<br />\[ x = \frac{-10 \pm \sqrt{100 + 300}}{10} \]<br />\[ x = \frac{-10 \pm \sqrt{400}}{10} \]<br />\[ x = \frac{-10 \pm 20}{10} \]<br /><br />So the roots are:<br />\[ x = \frac{10}{10} = 1 \]<br />\[ x = \frac{-30}{10} = -3 \]<br /><br />Thus, the factored form is:<br />\[ f(x) = 5(x - (-3))(x - 1) \]<br />\[ f(x) = 5(x + 3)(x - 1) \]<br /><br />## Step 3: List Key Properties<br />### The key properties of the function $f(x) = 5x^2 + 10x - 15$ are as follows:<br /><br />- **Vertex**: $(-1, -20)$<br />- **Axis of Symmetry**: $x = -1$<br />- **X-intercepts**: $x = -3$ and $x = 1$<br />- **Y-intercept**: $y = -15$