Problem 19 Graph each inequality. $$y \le... [FREE SOLUTION] (2024)

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Chapter 12: Problem 19

Graph each inequality. $$y \leq x^{2}-1$$

Short Answer

Expert verified

Graph the parabola $y = x^{2} - 1$ and shade the region below it including the parabola itself.

Step by step solution

Understand the inequality

The given inequality is: $y \leq x^{2}-1$. This means we are dealing with a parabola.

Graph the equation of the parabola

First, graph the equation $y = x^{2} - 1$. This will be the boundary of our inequality. It's a parabola that opens upwards with the vertex at (0, -1).

Determine the region for the inequality

Since the inequality is $y \leq x^{2}-1$, we include the area below the parabola as well as the parabola itself, because of the 'less than or equal to' part.

Shade the appropriate region

Shade the region below and including the parabola $y \leq x^{2}-1$. This represents all the possible values of $y$ that satisfy the inequality.

Key Concepts

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

parabola

In mathematics, a parabola is a U-shaped curve that can open either upwards or downwards. Parabolas are defined by quadratic equations in the form of $y = ax^2 + bx + c$. Here, we have the equation $y = x^2 - 1$. The coefficient of $x^2$ is positive, which means the parabola opens upwards. Every parabola has a special point called the vertex, which is the highest or lowest point on the graph. For our equation $y = x^2 - 1$, the vertex is at the point (0, -1), meaning that's where the vertex of the parabola is located along the y-axis. Parabolas are symmetric around a vertical line called the axis of symmetry. In this case, our axis of symmetry is the y-axis itself (x = 0). Knowing how to graph a parabola allows us to better understand and visualize inequalities involving them. It's the foundation for the next steps in solving and graphing inequalities.

inequality graphing

Understanding and graphing inequalities is closely tied to graphing equations. When we graph an inequality like $y \leq x^2 - 1$, the critical step is first to graph the corresponding equation $y = x^2 - 1$ as a boundary. This boundary is the graph of the parabola. The key difference in graphing an inequality versus an equation is that for inequalities, we're interested in regions of the coordinate plane that satisfy the inequality. To determine which side or region of this boundary satisfies the inequality, we need to test points from the plane. For instance, in $y \leq x^2 - 1$, once we graph the parabola, we know the inequality holds true for all points below or on the parabola. This is because the 'less than or equal to' part tells us that we're including the boundary (the parabola itself) and everything below it. Thus, the graph includes not just the curve but a whole region that fulfills the given condition in the inequality.

shading regions

Shading regions is a visual way to represent solutions for inequalities on a graph. After graphing the boundary, you need to identify the correct region to shade. For our inequality $y \leq x^2 - 1$, we know the boundary is the parabola $y = x^2 - 1$. Since the inequality is 'less than or equal to', we shade the region below the parabola. This visual technique is useful because it clearly shows all the points (x, y) that make the inequality true. Here are a few tips:

Always start by graphing the boundary first.
Check a point to determine which side of the boundary to shade; the origin (0,0) often works well unless it lies on the boundary.
For inequalities with 'less than or equal to' ($\leq$) or 'greater than or equal to' ($\geq$), use a solid line for the boundary. For strict inequalities ('less than' < or 'greater than' >), use a dashed line.
Shading correctly helps avoid mistakes and is especially important for complex systems of inequalities.

Shading regions makes it easier to visually understand and solve inequalities, especially when dealing with multiple inequalities.

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$Problem 19 Graph each inequality. $$y \le... [FREE SOLUTION] (3)$

Most popular questions from this chapter

Solve each system of equations. If the system has no solution, state that itis inconsistent. $$ \left\\{\begin{array}{l} 3 x-6 y=7 \\ 5 x-2 y=5 \end{array}\right. $$Based on material learned earlier in the course. The purpose of these problemsis to keep the material fresh in your mind so that you are better prepared forthe final exam. Find the area of the triangle with vertices at $(0,5),(3,9),$ and (12,0)Curve Fitting Find real numbers $a, b,$ and $c$ so that the graph of thefunction $y=a x^{2}+b x+c$ contains the points $(-1,-2),(1,-4),$ and (2,4)Verify that the values of the variables listed are solutions of the system ofequations. $$ \begin{array}{l} \left\\{\begin{aligned} 3 x+3 y+2 z &=4 \\ x-3 y+z &=10 \\ 5 x-2 y-3 z &=8 \end{aligned}\right. \\ x=2, y=-2, z=2 ;(2,-2,2) \end{array} $$Solve each system of equations. If the system has no solution, state that itis inconsistent. $$ \left\\{\begin{array}{l} \frac{1}{x}+\frac{1}{y}=8 \\ \frac{3}{x}-\frac{5}{y}=0 \end{array}\right. $$

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