Problem 29 - A Graphical Approach to Precalculus with Limits

Question

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-4 x+5$$

Step-by-Step Solution

Answer

The vertex is at (1, 3), and the parabola opens upwards.

Step 1: Identify Coefficients

The quadratic function is given by \( P(x) = 2x^2 - 4x + 5 \). Identify the coefficients \( a = 2 \), \( b = -4 \), and \( c = 5 \). These are necessary to use the vertex formula.

Step 2: Use the Vertex Formula

The vertex \((h, k)\) of a quadratic function in the form \( ax^2 + bx + c \) is found using the formula \( h = \frac{-b}{2a} \). Substitute the values: \( h = \frac{-(-4)}{2 \cdot 2} = \frac{4}{4} = 1 \).

Step 3: Find the Y-coordinate of Vertex

Substitute \( h = 1 \) into the original function to find \( k \). Thus, \( k = P(1) = 2(1)^2 - 4(1) + 5 = 2 - 4 + 5 = 3 \). Therefore, the coordinates of the vertex are \( (1, 3) \).

Step 4: Determine the Parabola's Direction

Since the coefficient \( a = 2 \) is positive, the parabola opens upwards. This gives information on the general shape of the graph.

Step 5: Identify Additional Points

Select a few values of \( x \) to find additional points for graphing. For example, calculate \( P(0) = 5 \) and \( P(2) = 5 \). These points are \( (0, 5) \) and \( (2, 5) \).

Step 6: Sketch the Graph

Plot the vertex \((1,3)\) and the additional points \((0, 5)\) and \((2, 5)\) on a coordinate plane. Draw a parabola through these points ensuring it opens upwards.

Key Concepts

Vertex Formula

The vertex of a quadratic function is a crucial point where the graph changes direction. Let's find it using the vertex formula.
The quadratic function given is in the form \( ax^2 + bx + c \), where the vertex \(( h, k )\) can be calculated using the formula \( h = \frac{-b}{2a} \).
This formula stems from the axis of symmetry of the parabola. It's essentially the line that divides the parabola into mirror-image halves, passing through the vertex. To apply it, plug in the values from the coefficients:

• \( a = 2 \)
• \( b = -4 \)

Use these to find \( h \). Substituting, we have:

• \( h = \frac{-(-4)}{2 \times 2} = \frac{4}{4} = 1 \)

Once \( h \) is known, find \( k \) by substituting \( h \) back into the original function \( P(x) \):

• \( k = P(1) = 2(1)^2 - 4(1) + 5 = 3 \)

Thus, the vertex is \(( 1, 3 )\). This tells us the parabola's highest or lowest point is at \(( 1, 3 )\). Knowing the vertex is important for both graphing and understanding the function's peak or trough.

Parabola Graphing

Graphing a parabola allows us to visualize the quadratic function. This involves plotting specific points and understanding the symmetry in the graph.
First, plot the vertex \((1, 3)\), which is the turning point of the parabola. Since we earlier identified that \( a = 2 \) is positive, we know the parabola opens upwards. An upward opening parabola indicates a minimum point at the vertex.
To graph effectively:

• Begin with the vertex as a reference.
• Plot additional points on both sides of the vertex to ensure accuracy and mirror symmetry.

In the original exercise, we selected \( x = 0 \) and \( x = 2 \):

• \( P(0) = 5 \), giving the point \((0, 5)\)
• \( P(2) = 5 \), giving the point \((2, 5)\)

Both these points happen to be at the same height, illustrating the symmetry about the line \( x = 1 \). With these points, draw a smooth curve through them. The parabola will visually confirm the understanding of the function's behavior, showing its upward-opening shape.

Coefficient Identification

Identifying the coefficients \( a \), \( b \), and \( c \) is the first step in working with quadratic functions. Each of these coefficients plays a significant role in determining the characteristics of the graph.
For our function \( P(x) = 2x^2 - 4x + 5 \):

• \( a = 2 \)
• \( b = -4 \)
• \( c = 5 \)

These values affect the parabola's shape and position:

• \( a \) determines whether the parabola opens upwards or downwards. A positive \( a \) means it opens upwards, while a negative \( a \) opens downwards.
• \( b \) impacts the location of the vertex along the x-axis. It is used in the vertex formula to find \( h \).
• \( c \) is the constant term and often indicates the y-intercept of the graph. Here, \( P(0) = 5 \) confirms the y-intercept.

Understanding these coefficients helps in predicting the graph's appearance and the function's behavior before graphing.