Problem 26 - Intermediate Algebra

Question

Graph each system. See Examples 3 and 4. $$ \left\\{\begin{array}{l} {y \leq-x^{2}+3} \\ {y \leq 2 x-1} \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the parabola and line, and shade below both; the solution is the overlap.

Step 1: Graph the First Inequality

The first inequality is \(y \leq -x^{2}+3\). Start by considering the equation \(y = -x^{2}+3\), which is a downward opening parabola. Graph this parabola in the \(xy\)-plane. Since the inequality is \(\leq\), shade the area below the parabola to indicate all the points \((x, y)\) that satisfy the inequality.

Step 2: Graph the Second Inequality

The second inequality is \(y \leq 2x - 1\). Start by considering the line \(y = 2x - 1\). This line can be graphed by finding its intercepts or using the slope-intercept form. Graph this line on the same \(xy\)-plane. Shade the area below the line, as the inequality indicates \(\leq\).

Step 3: Identify the Feasible Region

The solution to the system of inequalities is the region where the shaded areas overlap. Look for the intersection of the shaded regions from the parabola and the line. This overlapping region is the set of \((x, y)\) pairs that satisfy both inequalities simultaneously.

Key Concepts

Parabolas

Parabolas are a type of curve that is often found in quadratic equations. A typical quadratic equation is written in the form \[ y = ax^2 + bx + c \],where the graph of this function creates a U-shaped or inverted U-shaped curve depending on the value of \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

In the original inequality, \(y \leq -x^2 + 3\), the parabola opens downwards since the coefficient of \(x^2\) is negative. When graphing, it is crucial to plot the parabola correctly by selecting several points or by identifying the vertex and axis of symmetry.
By shading the area below the parabola, we represent all the solutions to the inequality, covering all possible points where \(y\) is less than or equal to values on the curve.

Linear Equations

Linear equations graph as straight lines and are generally represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. They represent relationships where a change in \(x\) results in a proportional change in \(y\).
In the inequality \(y \leq 2x - 1\),

• The slope, \(m = 2\), indicates a rise of 2 units for every 1 unit move to the right.
• The y-intercept, \(b = -1\), shows that the line crosses the y-axis at point (0, -1).

To graph this line, one may use the intercepts or plot points derived by inserting values into the equation. The shaded region below this line contains all the points satisfying the inequality. This demonstrates that for each \(x\), \(y\) is less than or equal to the line's corresponding value.

Feasible Region

The feasible region in a system of inequalities is the overlap where all inequalities are true simultaneously. In graphing terms, this is the area where the shaded regions of both inequalities intersect and represent possible solutions.
For the provided system, the feasible region is found at the intersection of:

• The downward opening parabola \(y \leq -x^2 + 3\), and
• The straight line \(y \leq 2x - 1\).

To identify the feasible region, observe where the shading from both the parabola and line overlap. This area signifies all \((x, y)\) pairs that satisfy both inequalities. It symbolizes the joint solutions, highlighting the realistic set of outcomes constrained by these equations. The feasible region is crucial for systems of inequalities as it showcases boundaries within a solution can be applied or sought in practical situations.