1. **State the problem:** We are given two functions: $$f(x) = 4x + 7$$ $$g(x) = -2x^2 + 11x + 1$$ We need to find which of the given points lies on both graphs, i.e., where $f(x) = g(x)$. 2. **Set the functions equal to find intersection points:** $$4x + 7 = -2x^2 + 11x + 1$$ 3. **Rearrange the equation to standard quadratic form:** $$0 = -2x^2 + 11x + 1 - 4x - 7$$ $$0 = -2x^2 + (11x - 4x) + (1 - 7)$$ $$0 = -2x^2 + 7x - 6$$ 4. **Multiply both sides by $-1$ to simplify:** $$0 = 2x^2 - 7x + 6$$ 5. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot 6}}{2 \cdot 2} = \frac{7 \pm \sqrt{49 - 48}}{4} = \frac{7 \pm 1}{4}$$ 6. **Calculate the two roots:** $$x_1 = \frac{7 + 1}{4} = \frac{8}{4} = 2$$ $$x_2 = \frac{7 - 1}{4} = \frac{6}{4} = 1.5$$ 7. **Find corresponding $y$ values for each $x$ using $f(x)$:** For $x=2$: $$f(2) = 4(2) + 7 = 8 + 7 = 15$$ For $x=1.5$: $$f(1.5) = 4(1.5) + 7 = 6 + 7 = 13$$ 8. **Check which given points match these $(x,y)$ pairs:** - Option B: $(1.5, 13)$ matches exactly. - Option C: $(2, 1.5)$ does not match since $y=15$ for $x=2$. - Options A and D do not match the $x$ values found. **Final answer:** The graphs intersect at point **B. (1 1/2, 13)**.