Problem 26 Sketch the graph of the inequali... [FREE SOLUTION]

Chapter 9: Problem 26

Sketch the graph of the inequality. $$y<-x^{2}-3 x-1$$

Short Answer

Expert verified

Vertex: (1.5, -2.25), x-intercepts: (-1,0), (1,0). A downward-opening parabola plotted using these points, with the region below the parabola shaded.

Step by step solution

Find the Vertex

The vertex of the parabola $y = ax^2 + bx + c$ is given by $(-b/(2a), f(-b/(2a))$. For the given equation $y = -x^{2} - 3x - 1$, a = -1, b = -3, so the vertex is $(-(-3)/(2*-1), -(-3)^2/(4*-1) - 3*(-3)/(2*-1) - 1) = (1.5, -2.25)$.

Find the x-intercepts

The x-intercepts are the roots of the equation $y = 0$. Set $y = -x^{2} - 3x - 1 = 0$ and solve for x using the quadratic formula $x = [-b卤sqrt(b^2-4ac)]/(2a)$, i.e., $x = [3卤sqrt((-3)^2-4*-1*-1)]/(2*-1)$. Therefore, the x-intercepts are $((-1,0), (1,0))$. The vertex and x-intercepts give us key points to locate the curve.

Draw the parabola

With the vertex and x-intercepts, and know that this is a downward opening parabola because the coefficient of $x^2$ is negative. Sketch the graph of the parabola.

Shade the region of inequality

The inequality is $y < -x^{2} - 3x - 1$, therefore, shade the region below the parabola. That portrayed area represents all points (x, y) that satisfy the inequality.

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Key Concepts

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

Parabola Vertex

The vertex of a parabola is a crucial point that helps us understand the shape and position of the curve. For any quadratic equation in the form of $y = ax^2 + bx + c$, the vertex can be found using the formula

$x = \frac{-b}{2a}$, which gives the x-coordinate of the vertex.
Substitute this x-value into the equation $y = ax^2 + bx + c$ to find the y-coordinate.

For the equation $y = -x^2 - 3x - 1$, by applying $a = -1$ and $b = -3$, we calculate the vertex as $(1.5, -2.25)$. This point is particularly important because it indicates the maximum or minimum point of the parabola.
The negative sign before $x^2$ tells us the parabola opens downwards, so this vertex is actually its highest point.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation, especially when solving for x-intercepts. An equation of the form $ax^2 + bx + c = 0$ can be solved using:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

This formula gives the x-values where the parabola will cross the x-axis.
In our example, $a = -1$, $b = -3$, and $c = -1$. Plugging these values into the formula, we find the x-intercepts to be at $(-1, 0)$ and $(1, 0)$.
The "+" and "-" signs in the formula indicate the two potential x-intercepts, depending on the parabola's specific path.

x-intercepts

The x-intercepts are the points where a graph crosses the x-axis, indicating where the value of y is zero. They are found by solving the quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula.
This process reveals points that are crucial for graphing because they help define the parabola's position in relation to the x-axis.

They show where the curve changes direction.
For our example, we derived the intercepts as $(-1,0)$ and $(1,0)$.

These points assist in drawing an accurate graph and provide context for visualizing how the parabola interacts with the coordinate plane.
Understanding x-intercepts allows us to visualize the curve's orientation and is vital for step-by-step graph sketching.

Inequality Solutions

Graphing inequalities involves more than simply drawing the parabola. You also need to consider which regions satisfy the inequality. In the inequality $y < -x^2 - 3x - 1$, we are looking for all points (x, y) where y is less than the parabola.
After sketching the parabola using the vertex and x-intercepts, shade the region under the curve.
By shading the area below the parabola, we effectively mark the solution set that satisfies the inequality.

This demonstrates clearly how the inequality creates a distinct area on the graph.

This process involves identifying all values underneath the curve, providing a visual representation of all solutions for the given inequality.
Identifying and shading this region is essential to solving inequality problems graphically.

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91影视

Sketch the graph of the inequality. $$y<-x^{2}-3 x-1$$

Short Answer

Step by step solution

Find the Vertex

Find the x-intercepts

Draw the parabola

Shade the region of inequality

Key Concepts

Parabola Vertex

Quadratic Formula

x-intercepts

Inequality Solutions

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