1. **Problem statement:** Given the functions $f(x) = -2\left(x + \frac{2}{5}\right)$ and $g(x) = \frac{2 - 5x}{4}$, find the zeros of $f$ and $g$, their intersection point, and the angle between the two functions. 2. **Formula and rules:** - Zero of a function is where $f(x) = 0$. - Intersection point satisfies $f(x) = g(x)$. - The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by: $$\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$ 3. **Find zeros:** - For $f(x)$: $$0 = -2\left(x + \frac{2}{5}\right) \Rightarrow x + \frac{2}{5} = 0 \Rightarrow x = -\frac{2}{5}$$ - For $g(x)$: $$0 = \frac{2 - 5x}{4} \Rightarrow 2 - 5x = 0 \Rightarrow 5x = 2 \Rightarrow x = \frac{2}{5}$$ 4. **Find intersection point:** Set $f(x) = g(x)$: $$-2\left(x + \frac{2}{5}\right) = \frac{2 - 5x}{4}$$ Multiply both sides by 4: $$4 \cdot -2\left(x + \frac{2}{5}\right) = 2 - 5x$$ $$-8\left(x + \frac{2}{5}\right) = 2 - 5x$$ Expand left side: $$-8x - \frac{16}{5} = 2 - 5x$$ Bring all terms to one side: $$-8x + 5x = 2 + \frac{16}{5}$$ $$-3x = 2 + \frac{16}{5} = \frac{10}{5} + \frac{16}{5} = \frac{26}{5}$$ Divide both sides by -3: $$x = \frac{\cancel{\frac{26}{5}}}{\cancel{-3}} = -\frac{26}{15}$$ Find $y$ by substituting $x$ into $f(x)$: $$f\left(-\frac{26}{15}\right) = -2\left(-\frac{26}{15} + \frac{2}{5}\right) = -2\left(-\frac{26}{15} + \frac{6}{15}\right) = -2\left(-\frac{20}{15}\right) = -2 \cdot -\frac{4}{3} = \frac{8}{3}$$ So intersection point is $\left(-\frac{26}{15}, \frac{8}{3}\right)$. 5. **Find slopes:** - $f(x) = -2\left(x + \frac{2}{5}\right) = -2x - \frac{4}{5}$, slope $m_f = -2$. - $g(x) = \frac{2 - 5x}{4} = \frac{2}{4} - \frac{5}{4}x = \frac{1}{2} - \frac{5}{4}x$, slope $m_g = -\frac{5}{4}$. 6. **Calculate angle between $f$ and $g$:** $$\tan(\theta) = \left| \frac{-2 - \left(-\frac{5}{4}\right)}{1 + (-2) \cdot \left(-\frac{5}{4}\right)} \right| = \left| \frac{-2 + \frac{5}{4}}{1 + \frac{10}{4}} \right| = \left| \frac{-\frac{8}{4} + \frac{5}{4}}{1 + \frac{10}{4}} \right| = \left| \frac{-\frac{3}{4}}{\frac{14}{4}} \right| = \frac{3}{14}$$ Angle $\theta = \arctan\left(\frac{3}{14}\right) \approx 12.2^\circ$. **Final answers for part 1:** - Zero of $f$: $x = -\frac{2}{5}$ - Zero of $g$: $x = \frac{2}{5}$ - Intersection point: $\left(-\frac{26}{15}, \frac{8}{3}\right)$ - Angle between $f$ and $g$: approximately $12.2^\circ$