Problem 48 - Algebra 2

Question

Graph each equation. $$ 2 x^{2}-y+20 x=-53 $$

Step-by-Step Solution

Answer

The graph of the equation \(2x^2 + 20x + 53 = y\) is a vertical parabola facing upwards, determined by the rearrangement of terms and values for the vertex and y-intercept.

Step 1: Reorganize equation

Firstly, rearrange the equation to achieve a common form for graphing. For this, we will add \(y\) and 53 to each side. This yields: \(2x^2 + 20x + 53 = y\).

Step 2: Recognize form

We determine the equation to be in the form of \(y = ax^2 + bx + c\), which is the form of a quadratic equation where \(a = 2\), \(b = 20\), and \(c = 53\). The parameters define the type of the parabola. The general shape of the graph will be a vertical parabola due to the squared term being \(x\) and coefficient 'a' being positive.

Step 3: Find the vertex

The vertex of the parabola can be found using the formula \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\). Here \(h\) represents x-coordinate and \(k\) represents y-coordinate of the vertex. Substitute \(a = 2\) and \(b = 20\) into the formulas to obtain the vertex.

Step 4: Find the y-intercept

The vertical or y-intercept of the parabola can be found by substituting \(x = 0\) into the equation. This will give the point at which the graph intercepts the y-axis.

Step 5: Graph the equation

Mark the vertex and the intercepts on the graph. Next, plot a few more points for a more accurate graph. As this is a quadratic function and the manifest of a parabola, it is a curved line. It faces upwards when the coefficient of \(x^2\) (which is a) is positive

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that has its highest power as the square (or second power) of the variable. The general form of a quadratic equation is given by: \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this form, 'a' must not be zero, as that would reduce the equation to a linear one. Quadratic equations are often used in various fields such as physics, engineering, and economics. They can model different types of curves, most commonly known as parabolas. Understanding the components of the quadratic equation, such as the coefficients \(a\), \(b\), and \(c\), is crucial in graphing and solving them. When \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards. The value of \(b\) influences the axis of symmetry of the parabola while \(c\) represents the y-intercept.

Vertex Formula

The vertex of a parabola is a critical point that represents its highest or lowest point depending on its orientation. For parabolas opening upwards or downwards, this point is either a minimum or a maximum, respectively. The vertex coordinates \((h, k)\) can be calculated using formulas derived from the coefficients of the quadratic equation:

• \(h = -\frac{b}{2a}\)
• \(k = c - \frac{b^2}{4a}\)

By substituting \(a = 2\) and \(b = 20\) into these formulas, the values of \(h\) and \(k\) give you the exact location of the vertex. The vertex is crucial in graphing because it helps in plotting the central axis of the parabola. Understanding where the vertex is placed allows one to determine how the parabola is positioned with respect to the x and y axes, ensuring a more accurate graph.

y-Intercept

The y-intercept of a graph is the point at which it crosses the y-axis. For quadratic equations, this is found by setting \(x = 0\) in the equation \(y = ax^2 + bx + c\). This simplifies to \(y = c\). Therefore, the y-intercept is typically the constant term \(c\) in the quadratic equation. In the example equation, substituting \(x = 0\) reveals the y-intercept directly as the constant \(c = 53\). This point is essential for graphing as it provides one of the initial anchor points for plotting the curve. Additionally, it gives insight into the initial value of the function when no other variable is present or when time or distance starts from zero in real-world applications.

Graphing Techniques

Graphing a quadratic equation involves several steps to ensure accuracy and understanding of its behavior. First, write the equation in standard form \(y = ax^2 + bx + c\). This helps in identifying key characteristics like the vertex, y-intercept, and axis of symmetry. Placing these points correctly is crucial. Once the vertex is calculated using the vertex formula, it becomes the central point on the graph.

• Plot the vertex.
• Locate and mark the y-intercept.
• Reflect the y-intercept across the vertex to maintain symmetry.

After this, plot a few additional points on either side of the vertex to determine the curve accurately. The accuracy of the graph improves with more plotted points as they define the parabola's path. Drawing a smooth curve through all the points, ensuring it appropriately curls outwards, completes the parabola on the graph. Understanding and applying these techniques can provide visual insight and effectively solve real-world quadratic problems.