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Chapter 9: Problem 42

Graph the solution set of each system of inequalities. $$\begin{aligned}&y \geq(x-2)^{2}+3\\\&y \leq-(x-1)^{2}+6\end{aligned}$$

Short Answer

Expert verified
Graph the parabola of each inequality and shade the appropriate regions; the intersection of the shaded areas is the solution.

Step by step solution

01

- Graph the First Inequality

Graph the equation for the first inequality: \(y = (x-2)^{2} + 3\). This is a parabola opening upwards with its vertex at (2,3). Shade the region above this parabola since the inequality is \(y \geq (x-2)^{2} + 3\).
02

- Graph the Second Inequality

Graph the equation for the second inequality: \(y = -(x-1)^{2} + 6\). This is a parabola opening downwards with its vertex at (1,6). Shade the region below this parabola since the inequality is \(y \leq -(x-1)^{2} + 6\).
03

- Identify the Intersection Region

Identify the region where the shaded area from the first inequality overlaps with the shaded area from the second inequality. This common area is the solution set to the system of inequalities.
04

- Verify Points in the Intersection Area

Pick a point within the intersection region and verify it satisfies both inequalities. For example, checking the point (2, 4) in both inequalities confirms it lies within the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parabolas
A parabola is a U-shaped graph that represents a quadratic equation. When graphing parabolas, it's essential to understand their vertex, direction, and width.
Here's a quick guide:
  • The equation of a parabola in vertex form is given by: \[ y = a(x-h)^{2} + k \]
    where (h, k) is the vertex.
  • If \[ a > 0 \], the parabola opens upward. If \[ a < 0 \], it opens downward.
  • The larger the absolute value of 'a,' the narrower the parabola. The smaller the absolute value of 'a,' the wider the parabola.
In the given problems:
  • \

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Most popular questions from this chapter

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

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Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}$$
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