1. **State the problem:** We need to express the function $g(x) = 2x^2 - 13x - 7$ in factored form, find its zeros, the equation of the axis of symmetry, and the coordinates of the vertex. 2. **Factoring the quadratic:** The function is $g(x) = 2x^2 - 13x - 7$. We look for two numbers that multiply to $2 \times (-7) = -14$ and add to $-13$. These numbers are $-14$ and $1$. Rewrite the middle term: $$2x^2 - 14x + x - 7$$ Group terms: $$ (2x^2 - 14x) + (x - 7) $$ Factor each group: $$ 2x(x - 7) + 1(x - 7) $$ Factor out common binomial: $$ (2x + 1)(x - 7) $$ 3. **Zeros of the function:** Set each factor equal to zero: $$ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} $$ $$ x - 7 = 0 \Rightarrow x = 7 $$ So, zeros are $x = -\frac{1}{2}$ and $x = 7$. 4. **Axis of symmetry:** The axis of symmetry is the vertical line halfway between the zeros: $$ x = \frac{-\frac{1}{2} + 7}{2} = \frac{\frac{-1}{2} + \frac{14}{2}}{2} = \frac{\frac{13}{2}}{2} = \frac{13}{4} $$ 5. **Coordinates of the vertex:** The vertex lies on the axis of symmetry. Find $g\left(\frac{13}{4}\right)$: $$ g\left(\frac{13}{4}\right) = 2\left(\frac{13}{4}\right)^2 - 13\left(\frac{13}{4}\right) - 7 $$ Calculate step-by-step: $$ 2 \times \frac{169}{16} - \frac{169}{4} - 7 = \frac{338}{16} - \frac{169}{4} - 7 $$ Simplify fractions: $$ \frac{338}{16} - \frac{676}{16} - \frac{112}{16} = \frac{338 - 676 - 112}{16} = \frac{-450}{16} = -\frac{225}{8} $$ So, vertex coordinates are $\left(\frac{13}{4}, -\frac{225}{8}\right)$. **Final answers:** - Factored form: $g(x) = (2x + 1)(x - 7)$ - Zeros: $x = -\frac{1}{2}, 7$ - Axis of symmetry: $x = \frac{13}{4}$ - Vertex: $\left(\frac{13}{4}, -\frac{225}{8}\right)$