Problem 29 - A Graphical Approach to College Algebra

Question

For each quadratic function defined , (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 of the function is (1, 3).

Step 1: Identify the Quadratic Function

Given the quadratic function \( P(x) = 2x^2 - 4x + 5 \), we identify it as a standard form quadratic equation: \( ax^2 + bx + c \). Here, \( a = 2 \), \( b = -4 \), and \( c = 5 \).

Step 2: Apply the Vertex Formula

The vertex \((h, k)\) of a quadratic function in the form \(ax^2 + bx + c\) can be found using the formula \( h = -\frac{b}{2a} \). Substitute \(a = 2\) and \(b = -4\) into the formula:\[ h = -\frac{-4}{2(2)} = \frac{4}{4} = 1 \]

Step 3: Calculate the Vertex's y-coordinate

To find \( k \), the y-coordinate of the vertex, substitute \( h = 1 \) back into the function \( P(x) \):\[ k = P(1) = 2(1)^2 - 4(1) + 5 = 2 - 4 + 5 = 3 \] Thus, the vertex is \((1, 3)\).

Step 4: Graph the Function

To graph the quadratic function, plot the vertex \((1, 3)\). Since the coefficient \( a = 2 \) is positive, the parabola opens upwards. Determine additional points symmetrically around the vertex, compute their y-values, and draw the parabola using these points and symmetry.

Key Concepts

Vertex Formula

Finding the vertex of a quadratic function is a key part of understanding its graph. A quadratic function can be written in standard form as \( ax^2 + bx + c \). The formula to find the vertex \((h, k)\) is very helpful and is given by:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

The vertex \((h, k)\) is the point where the parabola changes direction, and it's either the highest or lowest point on the graph depending on the nature of the parabola.
In our example, the function \( P(x) = 2x^2 - 4x + 5 \) has coefficients \( a = 2 \) and \( b = -4 \). Using the vertex formula, we substitute these values to find:

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

Once we have \( h \), plug it back into the function to find \( k \):

• \( k = P(1) = 3 \)

Thus, our vertex is at \((1, 3)\). This represents the highest or lowest point of the parabola, which is crucial for understanding the graph's shape.

Graphing Parabolas

Graphing a quadratic function involves visualizing the "U" shape of a parabola. This is an important skill, as the graph gives insight into the function's properties and behaviors.

Start by plotting the vertex you found using the vertex formula. In this case, the vertex \((1, 3)\) is plotted on a coordinate plane.
Next, observe the coefficient of \(x^2\), known as \(a\). When \(a\) is positive, like \(a = 2\) in our example, the parabola opens upwards. If \(a\) were negative, it would open downwards.

To fully graph the parabola, find additional points symmetrically around the vertex. For example:

• Choose \(x = 0\), calculate \(P(0) = 5\), so plot \((0, 5)\)
• Choose \(x = 2\), calculate \(P(2) = 5\), so plot \((2, 5)\)

These points show symmetry around the vertex. Connect the points to form the U-shape of the parabola, ensuring it's symmetric. The graphed parabola opens upwards, showing that the vertex is at the minimum point.

Quadratic Equation Standard Form

The standard form of a quadratic equation is vital for performing calculations such as finding the vertex and graphing the function. The standard form is written as \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\).

This form is beneficial because:

• It directly allows us to determine the direction of the parabola based on the sign of \(a\).
• It provides the coefficients necessary to use the vertex formula.
• It simplifies substitution to calculate specific values of the function.

In our given function \( P(x) = 2x^2 - 4x + 5 \), we quickly identified it as \( ax^2 + bx + c \) with \(a = 2\), \(b = -4\), and \(c = 5\). Knowing this form makes it easier to perform various operations, including completing the square or using the quadratic formula to find roots or intercepts. Understanding this foundational form allows us to navigate the function's properties and enhances our ability to visualize and compute exact points.