Problem 31 - Algebra 1

Question

Sketch the graph of the inequality. $$y \geq 2 x^{2}+5 x+3$$

Step-by-Step Solution

Answer

The graph of the given inequality is an upwards opening parabola with a solid line. The area above the parabola, including the parabola itself, is shaded to represent the solution region for the inequality.

Step 1: Identify the type of parabola and its key points

The given equation is of a quadratic function, which is in the form \(y = ax^{2} + bx + c\). Therefore, it can be inferred it will be a parabola. The vertex of the parabola can be obtained by the formula \(-b/2a\), which would give the x-coordinate of the vertex. Substituting this x-value in the given equation will produce the y-coordinate.

Step 2: Determine the shape of the parabola

The leading coefficient (2 in this case) is greater than zero, therefore the parabola will open upwards.

Step 3: Plot the parabola and the required region

Draw the parabola graph using the identified key points and inferred shape. Since the inequality is '≥', the line of the graph should be solid indicating that points on the line are part of the solution. Moreover, you should shade the area above the curve to represent the solutions for the given inequality.

Key Concepts

Graphing Parabolas

Graphing a parabola involves presenting a U-shaped curve that can either open upwards or downwards, based on the quadratic equation. To effectively graph a parabola, there are few key aspects to consider.

• Firstly, recognize it is a quadratic function in the form: \(y = ax^2 + bx + c\).
• The leading coefficient \(a\) dictates the direction. If \(a > 0\), the parabola opens upward. If \(a < 0\), it opens downward.
• The vertex serves as the peak or lowest point, making it crucial for accurate graphing.
• Axis of symmetry is a vertical line through the vertex, helpful in ensuring the parabola is symmetric.

Once these key points are identified, plot the vertex and a few other points on either side. Draw the parabola using these points, ensuring it illustrates direction and location accurately. For inequalities like \(y \geq 2x^2 + 5x + 3\), shade the region above the curve to include solutions where the inequality holds true.

Vertex Formula

The vertex formula is a critical tool for finding the vertex of a parabola with ease. The vertex is the turning point of a parabola and is found using the formula for the x-coordinate:

\[x = -\frac{b}{2a}\]For the quadratic expression \(ax^2 + bx + c\), this formula gives you the x-value of the vertex. Substitute this x-value back into the original quadratic equation to find the y-coordinate of the vertex. This combined value \((x, y)\) becomes the vertex.

Understanding the vertex is crucial as it helps in sketching the parabola accurately on a coordinate plane. It also identifies whether the parabola is at a minimum or maximum point. In problems involving inequalities, the vertex can help determine which region to shade, indicating which side contains possible solutions.

Quadratic Functions

Quadratic functions are polynomial functions represented by the equation:

\[y = ax^2 + bx + c\]These functions characterize a parabolic graph – a smooth, symmetrical curve. Quadratics are central to algebra and appear in various real-world contexts from physics to finance. Key features distinguishing quadratic functions include:

• The 'a' value determines if the curve opens upward or downward.
• The vertex, calculated by \(-b/2a\), provides the pivotal point of the parabola.
• The axis of symmetry, a vertical line through the vertex, divides the parabola into two mirrored halves.

Solving quadratic inequalities involves sketching this parabola and identifying the regions where the inequality holds. The graph itself represents set values for \(x\) and \(y\) that satisfy the equation and inequality rules. Understanding how to manipulate these functions allows for flexibility in solving complex quadratic problems.