Problem 26 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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 1: 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 \((-(-3)/(2*-1), -(-3)^2/(4*-1) - 3*(-3)/(2*-1) - 1) = (1.5, -2.25)\).

Step 2: 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 \(x = [-b±sqrt(b^2-4ac)]/(2a)\), i.e., \(x = [3±sqrt((-3)^2-4*-1*-1)]/(2*-1)\). Therefore, the x-intercepts are \(((-1,0), (1,0))\). The vertex and x-intercepts give us key points to locate the curve.

Step 3: 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.

Step 4: 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.

Key Concepts

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.