Problem 35 - Contemporary Precalculus

Question

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$y=4(x-1)^{2}+2$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the equation $$y = 4(x - 1)^2 + 2$$ and label the vertex. Answer: The graph of the equation is a U-shaped parabola opening upwards with vertex at the point (1, 2).

Step 1: Identify the vertex

The given equation is $$y = 4(x - 1)^2 + 2$$. From this equation, we can see that the vertex is given by the point (h, k) = (1, 2).

Step 2: Identify the direction of the parabola

Since the value of a is 4, which is positive, the parabola will open upwards. This means that the parabola will be U-shaped, with the vertex being the lowest point on the graph.

Step 3: Plot the vertex

Begin by plotting the vertex (1, 2) on the coordinate plane. This will be our starting point for the graph.

Step 4: Determine additional points to plot

Let's find two more points on either side of the vertex to get a better idea of the shape of the parabola. To find these points, choose two x-values on either side of the vertex's x-coordinate (x = 1). We will use x = 0 and x = 2. 1. For x = 0: $$y = 4(0 - 1)^2 + 2 = 4 + 2 = 6$$ 2. For x = 2: $$y = 4(2 - 1)^2 + 2 = 4 + 2 = 6$$ The points (0, 6) and (2, 6) are both on the parabola.

Step 5: Plot the additional points

Plot the points (0, 6) and (2, 6) on the coordinate plane, along with the vertex (1, 2).

Step 6: Sketch the parabola

Connect the points (0, 6), (1, 2), and (2, 6) with a smooth U-shaped curve. This is the graph of the equation $$y = 4(x - 1)^2 + 2$$.

Step 7: Label the vertex

Label the vertex (1, 2) on the graph to complete the sketch.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the general form:\[ y = ax^2 + bx + c \]where:

• \(a\), \(b\), and \(c\) are constants,
• \(x\) is the variable, and
• \(a eq 0\) since this would make the equation linear, not quadratic.

They are used to describe curves known as parabolas, which have a symmetrical U-shape. The solutions to these equations, or the values of \(x\) that satisfy the equation, can be found using methods like factoring, completing the square, or the quadratic formula. Quadratic equations have two roots, which may be real or complex numbers, and these roots are where the parabola intersects the x-axis. Quadratic equations are pivotal in various fields such as physics, engineering, and in solving real-world problems where relationships can be represented as parabolas.

Vertex of a Parabola

The vertex of a parabola is a crucial point, as it indicates the highest or lowest point on the graph. For a quadratic equation in vertex form:\[ y = a(x-h)^2 + k \]The coordinates \((h, k)\) represent the vertex of the parabola. Whether the vertex represents a maximum or minimum value depends on the sign of \(a\):

• If \(a\) is positive, the parabola opens upwards, making the vertex the minimum point.
• If \(a\) is negative, the parabola opens downwards, with the vertex as the maximum point.

Finding the vertex is the first step in sketching the graph of a parabola, as it provides a reference point for plotting the entire curve. In our example, the vertex is at \((1, 2)\), which is the lowest point of the parabola since \(a = 4\), and is positive.

Graphing Parabolas

Graphing parabolas involves plotting points and understanding their symmetry and shape. The vertex form of a quadratic equation makes this process more accessible. Start by plotting the vertex, our central point, and then find additional points around it. This can be done by selecting x-values near the vertex and solving for \(y\). For example, in the given equation \(y = 4(x-1)^2 + 2\), points like \((0, 6)\) and \((2, 6)\) are equidistant from the vertex, which shows the symmetry of the parabola. After plotting these points:

• Draw a smooth curve through them, maintaining the symmetry centered around the vertex.
• Ensure the parabola opens in the correct direction based on the value of \(a\).
• Label all critical points such as the vertex and additional plotted points.

Graphing parabolas visually demonstrates the nature of quadratic equations and helps us analyze their real-world applications more effectively.