Problem 15 - A Graphical Approach to College Algebra

Question

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+4 x-1$$

Step-by-Step Solution

Answer

(a) The function in vertex form is \( P(x) = 3(x+\frac{2}{3})^2 - \frac{7}{3} \). (b) The vertex is \( (-\frac{2}{3}, -\frac{7}{3}) \). (c) Graph opens upwards at the vertex.

Step 1: Identify the Form to Convert

We need to rewrite the quadratic function \( P(x) = 3x^2 + 4x - 1 \) in the form \( P(x) = a(x-h)^2 + k \). This involves completing the square.

Step 2: Complete the Square

For the quadratic \( P(x) = 3x^2 + 4x - 1 \), factor out 3 from the first two terms: \[ P(x) = 3(x^2 + \frac{4}{3}x) - 1. \] Next, complete the square for the expression inside the parentheses. Take half of the coefficient of \( x \), square it, and add and subtract it inside the bracket: \[ x^2 + \frac{4}{3}x + \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^2 = \left(x + \frac{2}{3}\right)^2 - \frac{4}{9}. \] Substitute this back:\[ P(x) = 3 \left((x + \frac{2}{3})^2 - \frac{4}{9}\right) - 1 = 3(x + \frac{2}{3})^2 - \frac{4}{3} - 1. \] Finally, simplify the expression:\[ P(x) = 3(x + \frac{2}{3})^2 - \frac{7}{3}. \]

Step 3: Identify the Vertex Form Parameters

From the expression \( P(x) = 3(x+\frac{2}{3})^2 - \frac{7}{3} \), it is clear that \( h = -\frac{2}{3} \), \( k = -\frac{7}{3} \), and \( a = 3 \). The vertex of the parabola is \( \left(-\frac{2}{3}, -\frac{7}{3}\right) \).

Step 4: Sketching the Graph

To sketch the graph, note that \( a = 3 \) indicates the parabola opens upwards and is narrower than the standard parabola \( y = x^2 \) because \( a > 1 \). Plot the vertex at \( \left(-\frac{2}{3}, -\frac{7}{3} \right) \), and sketch the parabola opening upwards while maintaining symmetry about the vertical line \( x = -\frac{2}{3} \).

Key Concepts

Completing the Square

Completing the square is a useful algebraic technique that allows us to transform a standard quadratic function into a form that reveals important features, such as the vertex of a parabola. This technique involves taking a quadratic expression of the form \( ax^2 + bx + c \) and rewriting it in the form \( a(x-h)^2+k \). Here’s how we apply it:

• First, factor out the coefficient of \( x^2 \) (if it is not 1) from the first two terms of the quadratic expression.
• Next, identify the coefficient of the linear term (the \( x \) term), divide it by 2, and square it. This value is what you add and subtract inside the expression to complete the square.
• Adjust the constant term ('\( k \)') outside the bracket to maintain the balance of the equation.

By following these steps, you transform the original equation into a perfect square trinomial, which elegantly reveals the vertex and the opening direction of the parabola.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is particularly insightful for graphing and understanding parabolas. This form is given by \( P(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola and \( a \) affects the width and direction of the parabola.

• Vertex: The vertex form makes it easy to read the vertex directly from the equation, without any further calculations. If the equation is in vertex form, \( h \) and \( k \) are simply the x and y coordinates, respectively, of the vertex.
• Direction of Opening: The sign of \( a \) determines whether the parabola opens upwards or downwards. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• Width of the Parabola: The absolute value of \( a \) impacts the width. The greater the absolute value of \( a \), the narrower the parabola; a smaller absolute value means the parabola is wider.

Understanding the vertex form helps significantly in graphing and analyzing the nature of the quadratic function.

Graphing Parabolas

Graphing parabolas is a straightforward process once you understand the vertex form and the implications of its parameters. To graph a parabola:

• Locate the Vertex: Using the vertex coordinates \( (h, k) \), plot this crucial point on the graph.
• Determine the Axis of Symmetry: This vertical line runs through the vertex, while offering symmetry to the parabola. Its equation is \( x = h \).
• Identify the Direction and Width: Check the value of \( a \) to know which direction the parabola opens and its narrowness or wideness.
• Plot Additional Points: Choose x-values on either side of the vertex to find corresponding y-values, helping outline the parabola's shape.

By analyzing these elements, sketching a clear graph of the quadratic function is a simple task. This visualization aids in better understanding the behavior of quadratic functions.