1. **State the problem:** We are given the quadratic function $f(x) = 3x^2 + 5x - 7$. 2. **Express $f(x)$ in the form $a(x+h)^2 + k$ (completing the square):** - Start with $f(x) = 3x^2 + 5x - 7$. - Factor out the coefficient of $x^2$ from the first two terms: $$f(x) = 3\left(x^2 + \frac{5}{3}x\right) - 7$$ - To complete the square inside the parentheses, take half of the coefficient of $x$, which is $\frac{5}{3}$, so half is $\frac{5}{6}$. - Square this half: $$\left(\frac{5}{6}\right)^2 = \frac{25}{36}$$ - Add and subtract this inside the parentheses: $$f(x) = 3\left(x^2 + \frac{5}{3}x + \frac{25}{36} - \frac{25}{36}\right) - 7$$ - Group the perfect square trinomial and simplify: $$f(x) = 3\left(\left(x + \frac{5}{6}\right)^2 - \frac{25}{36}\right) - 7$$ - Distribute the 3: $$f(x) = 3\left(x + \frac{5}{6}\right)^2 - 3 \times \frac{25}{36} - 7 = 3\left(x + \frac{5}{6}\right)^2 - \frac{75}{36} - 7$$ - Simplify constants: $$- \frac{75}{36} - 7 = - \frac{75}{36} - \frac{252}{36} = - \frac{327}{36} = - \frac{109}{12}$$ - So the vertex form is: $$f(x) = 3\left(x + \frac{5}{6}\right)^2 - \frac{109}{12}$$ 3. **Solve the equation $f(x) = 0$:** - Set the vertex form equal to zero: $$3\left(x + \frac{5}{6}\right)^2 - \frac{109}{12} = 0$$ - Add $\frac{109}{12}$ to both sides: $$3\left(x + \frac{5}{6}\right)^2 = \frac{109}{12}$$ - Divide both sides by 3: $$\left(x + \frac{5}{6}\right)^2 = \frac{109}{36}$$ - Take the square root of both sides: $$x + \frac{5}{6} = \pm \sqrt{\frac{109}{36}} = \pm \frac{\sqrt{109}}{6}$$ - Solve for $x$: $$x = -\frac{5}{6} \pm \frac{\sqrt{109}}{6} = \frac{-5 \pm \sqrt{109}}{6}$$ 4. **Sketch the graph of $f(x) = 3x^2 + 5x - 7$:** - The vertex is at $$\left(-\frac{5}{6}, -\frac{109}{12}\right)$$. - The y-intercept is found by evaluating $f(0)$: $$f(0) = -7$$, so the point is $(0, -7)$. - The x-intercepts are the solutions to $f(x) = 0$, which are $$x = \frac{-5 + \sqrt{109}}{6}$$ and $$x = \frac{-5 - \sqrt{109}}{6}$$. The parabola opens upwards because the coefficient of $x^2$ is positive (3). Final answers: $$f(x) = 3\left(x + \frac{5}{6}\right)^2 - \frac{109}{12}$$ $$x = \frac{-5 \pm \sqrt{109}}{6}$$