Abs Tent Graph Fdb3Cd

1. **Problem statement:** We are given the function $$y = 5 - |x - 2|$$ and need to: (a) Sketch its graph. (b) Find the values of $x$ where $y < 0$. (c) Find the area bounded by the graph and the x-axis. 2. **Understanding the function:** The function is a vertical shift of the absolute value function $|x - 2|$ reflected and shifted up by 5. The graph of $y = |x - 2|$ is a V-shape with vertex at $(2,0)$. So $y = 5 - |x - 2|$ flips it upside down and shifts it up by 5. 3. **Sketching the graph:** - The vertex is at $x=2$, where $|x-2|=0$, so $y=5$. - For $x < 2$, $|x-2| = 2 - x$, so $y = 5 - (2 - x) = 3 + x$ (a line with slope 1). - For $x > 2$, $|x-2| = x - 2$, so $y = 5 - (x - 2) = 7 - x$ (a line with slope -1). The graph is a "tent" shape with peak at $(2,5)$ and intercepts where $y=0$. 4. **Finding x-intercepts (where $y=0$):** Set $5 - |x - 2| = 0$: $$|x - 2| = 5$$ This gives two solutions: $$x - 2 = 5 \implies x = 7$$ $$x - 2 = -5 \implies x = -3$$ So the graph crosses the x-axis at $x = -3$ and $x = 7$. 5. **Finding where $y$ is negative:** We want $5 - |x - 2| < 0$: $$|x - 2| > 5$$ This means $x$ is more than 5 units away from 2: $$x < -3 \quad \text{or} \quad x > 7$$ 6. **Finding the area bounded by the graph and the x-axis:** The graph is above the x-axis between $x = -3$ and $x = 7$. Area = integral of $y$ from $-3$ to $7$: Split integral at $x=2$: $$\text{Area} = \int_{-3}^{2} (5 - |x - 2|) dx + \int_{2}^{7} (5 - |x - 2|) dx$$ For $x < 2$, $|x - 2| = 2 - x$, so: $$\int_{-3}^{2} (5 - (2 - x)) dx = \int_{-3}^{2} (3 + x) dx$$ For $x > 2$, $|x - 2| = x - 2$, so: $$\int_{2}^{7} (5 - (x - 2)) dx = \int_{2}^{7} (7 - x) dx$$ Calculate each: $$\int_{-3}^{2} (3 + x) dx = \left[3x + \frac{x^2}{2}\right]_{-3}^{2} = \left(3(2) + \frac{2^2}{2}\right) - \left(3(-3) + \frac{(-3)^2}{2}\right) = (6 + 2) - (-9 + 4.5) = 8 - (-4.5) = 12.5$$ $$\int_{2}^{7} (7 - x) dx = \left[7x - \frac{x^2}{2}\right]_{2}^{7} = \left(7(7) - \frac{7^2}{2}\right) - \left(7(2) - \frac{2^2}{2}\right) = (49 - 24.5) - (14 - 2) = 24.5 - 12 = 12.5$$ Total area: $$12.5 + 12.5 = 25$$ **Final answers:** (a) The graph is a tent shape with vertex at $(2,5)$ and x-intercepts at $x = -3$ and $x = 7$. (b) $y$ is negative for $x < -3$ or $x > 7$. (c) The area bounded by the graph and the x-axis is $25$ square units.