Problem 19 - Intermediate Algebra

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. $$y=x^{2}+10 x+20$$

Step-by-Step Solution

Answer

The vertex of the parabola is \((-5, -5)\).

Step 1: Identify the Coefficients

The given quadratic equation is \( y = x^2 + 10x + 20 \). Here, the coefficients are \( a = 1 \), \( b = 10 \), and \( c = 20 \).

Step 2: Use the Vertex Formula

For a quadratic equation of the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex is found using the formula \( x = -\frac{b}{2a} \). Substitute \( b = 10 \) and \( a = 1 \) to get \( x = -\frac{10}{2 \times 1} = -5 \).

Step 3: Calculate the y-coordinate

Substitute \( x = -5 \) back into the equation to find the y-coordinate of the vertex: \( y = (-5)^2 + 10(-5) + 20 = 25 - 50 + 20 = -5 \).

Step 4: State the Vertex

The vertex of the parabola is \((-5, -5)\).

Step 5: Graph the Parabola

Plot the vertex \((-5, -5)\) on the coordinate plane. Then sketch the parabola using the standard shape of a parabola that opens upwards, as the coefficient of \( x^2 \) is positive.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable, usually denoted as \( x \), is squared. These equations fit the general form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. If \( a eq 0 \), then the equation represents a parabola when graphed.

The nature of quadratic equations makes them foundational in algebra. They are everywhere, from physics to finance, as they model situations where there is a squared relationship between variables. For example, the quadratic equation \( y = x^2 + 10x + 20 \) features a parabolic curve on a graph due to the \( x^2 \) term.

• **Coefficient \( a \)**: Determines the parabola's direction and width. Positive \( a \) means the parabola opens upwards.
• **Coefficient \( b \)**: Affects the position of the vertex along the x-axis.
• **Coefficient \( c \)**: The y-intercept where the parabola intersects the y-axis when \( x = 0 \).

Understanding and solving quadratic equations involve several techniques, including factoring, using the quadratic formula, and completing the square. However, when determining the vertex of the parabola, the vertex formula is most helpful.

Graphing Parabolas

Graphing parabolas involves plotting the curve of a quadratic equation, which typically results in a U-shaped graph. Knowing how to graph a parabola helps visualize the solution and understand the relationships within the quadratic equation.

To properly graph a parabola, follow these steps:

• **Find the vertex**: Use the vertex formula to determine the most significant point of the graph, which is the "turning point." The vertex for \( y = x^2 + 10x + 20 \) is \((-5, -5)\).
• **Plot the vertex**: Start by marking the vertex on the coordinate plane.
• **Determine the direction**: The sign of \( a \) (in this case, \( 1 \)) tells us the parabola opens upwards.
• **Draw the axis of symmetry**: This is a vertical line through the vertex at \( x = -5 \), dividing the parabola into two symmetrical halves.
• **Plot additional points**: Choose x-values around the vertex, substitute them into the equation, and find y-values to get more points for accurate sketching.
• **Draw the parabola**: Connect the points to form a smooth, U-shaped curve.

Graphing is crucial because it gives us a visual representation of the solutions to the equation and the nature of relationships within the parabola.

Vertex Formula

The vertex formula is a valuable tool in the field of algebra, especially when dealing with quadratic equations. It allows us to find the vertex of the parabola directly from the equation without needing to graph first.

For a quadratic equation in the form \( y = ax^2 + bx + c \), the vertex \((h, k)\) can be found using:

• **\( h = -\frac{b}{2a} \)**: This part of the formula gives the x-coordinate of the vertex. By substituting the values of \( b \) and \( a \) from the equation, we can find \( h \), the first part of our vertex coordinate.
• **\( k \):** Once \( h \) is calculated, substitute \( x = h \) back into the original quadratic equation to find the y-coordinate \( k \).

In our example, substituting \( b = 10 \) and \( a = 1 \) into \( x = -\frac{b}{2a} \) gives \( x = -5 \). Substituting \( x = -5 \) back into the original equation yields the y-coordinate as \( y = -5 \). Therefore, the vertex of the parabola \( y = x^2 + 10x + 20 \) is \((-5, -5)\).

The vertex formula is particularly useful because it quickly gives the highest or lowest point on the parabola, depending on its orientation. This point is crucial for understanding the maximum or minimum values in various applications.