1. **State the problem:** We want to find the formula for a trigonometric function of the form $$h(x) = a \cos(bx + c) + d$$ given that the function has a maximum at $\left(\frac{\pi}{2}, 5\right)$ and a minimum at $\left(\frac{\pi}{4}, -4\right)$. 2. **Identify amplitude $a$ and vertical shift $d$:** The amplitude is half the distance between max and min: $$a = \frac{5 - (-4)}{2} = \frac{9}{2} = 4.5$$ The vertical shift $d$ is the midpoint of max and min values: $$d = \frac{5 + (-4)}{2} = \frac{1}{2} = 0.5$$ 3. **Use max point to find phase shift:** At maximum, cosine equals 1, so inside the cosine must be zero: $$b \cdot \frac{\pi}{2} + c = 0$$ 4. **Use min point to relate $b$ and $c$:** At minimum, cosine equals -1, so inside cosine must be $\pi$ radians from max: $$b \cdot \frac{\pi}{4} + c = \pi$$ 5. **Set up system:** From max: $$c = -b \frac{\pi}{2}$$ Substitute into min: $$b \frac{\pi}{4} - b \frac{\pi}{2} = \pi$$ Simplify: $$b \left(\frac{\pi}{4} - \frac{\pi}{2}\right) = \pi$$ $$b \left(- \frac{\pi}{4}\right) = \pi$$ $$-\frac{b \pi}{4} = \pi$$ Multiply both sides by -4: $$b \pi = -4 \pi$$ Divide both sides by $\pi$: $$b = -4$$ 6. **Find $c$ using $b$:** $$c = -b \frac{\pi}{2} = -(-4) \frac{\pi}{2} = 2 \pi$$ 7. **Write final formula:** $$h(x) = 4.5 \cos(-4x + 2\pi) + 0.5$$ Since cosine is even: $$h(x) = 4.5 \cos(4x - 2\pi) + 0.5$$ This is the exact expression for $h(x)$ that matches the given max and min points.