Question

Complete the square of each quadratic expression. Then graph each function using graphing techniques. $$ f(x)=2 x^{2}-12 x+19 $$

Short answer

The function in vertex form is \( f(x) = 2(x - 3)^{2} + 1 \) with vertex \((3, 1)\).

Step-by-step solution

Identify the Quadratic Coefficient

Given the quadratic expression is in the form: \[ f(x) = ax^{2} + bx + c \] where \(a = 2\), \(b = -12\), and \(c = 19\). The first step is to identify these coefficients.

Factor Out the Leading Coefficient from the First Two Terms

Factor out the coefficient of the \(x^{2}\) term (which is 2) from the first two terms in the expression: \[ f(x) = 2(x^{2} - 6x) + 19 \].

Complete the Square

To complete the square, take half of the coefficient of \(x\) (which is -6), square it, and add and subtract this value inside the parentheses: \[ x^{2} - 6x \rightarrow x^{2} - 6x + 9 - 9 \]. This results in: \[ f(x) = 2(x^{2} - 6x + 9 - 9) + 19 \] or \[ f(x) = 2((x - 3)^{2} - 9) + 19 \].

Simplify the Expression

Distribute the 2 and combine like terms: \[ f(x) = 2(x - 3)^{2} - 18 + 19 \]. This simplifies to: \[ f(x) = 2(x - 3)^{2} + 1 \]. The quadratic is now in the form \(a(x - h)^{2} + k\), which makes it easier to graph.

Determine the Vertex and Graph the Function

The vertex form of the quadratic equation \(f(x) = 2(x - 3)^{2} + 1\) reveals the vertex \((h, k) = (3, 1)\). The coefficient 2 indicates a vertical stretch. Hence, the parabola opens upwards with the vertex at (3, 1). Plot the vertex and additional points, then draw the parabola to graph the function.

Key concepts

Quadratic Expression

A quadratic expression is a polynomial of degree 2. This means the highest power of the variable (usually x) is 2. In general form, a quadratic expression is written as: \[ ax^{2} + bx + c \] where:

• a, b, and c are constants
• x is the variable

In our exercise, the quadratic expression is given as: \[ f(x) = 2x^{2} - 12x + 19 \] Here, we need to complete the square to make it easier to graph.

Vertex Form

The vertex form of a quadratic expression is especially useful for graphing. The vertex form is given by: \[ f(x) = a(x - h)^{2} + k \] where:

• (h, k) is the vertex of the parabola
• a determines the 'vertical stretch' or 'compression'

Transforming the given quadratic expression into this form involves completing the square. In our example, after completing the square, we transform: \[ f(x) = 2x^{2} - 12x + 19 \] into: \[f(x) = 2(x - 3)^{2} + 1 \] In this new quadratic form, it becomes immediately clear that the vertex of the parabola is (3, 1).

Graphing Quadratic Functions

Graphing quadratic functions becomes simpler once you have the function in vertex form. The steps for graphing are as follows:

• Identify the vertex (h, k)
• Determine the direction of the parabola (upwards if a > 0, downwards if a < 0)
• Identify the vertical stretch or compression

In our example, \[ f(x) = 2(x - 3)^{2} + 1 \] you plot the vertex \((3, 1)\) and identify that the parabola opens upwards due to the positive coefficient 2. Plot additional points around the vertex to sketch the parabola's shape accurately.

Parabola Vertex

The vertex of a parabola gives you the highest or lowest point, depending on the direction it opens. It’s a crucial point in graphing as it helps to understand the shape and position. From the vertex form \[ f(x) = a(x - h)^{2} + k \], the vertex is simply given by:

• \[ h \] is the value that makes \[ (x - h)^{2} \] zero
• \[ k \] is the value of the function at \[ x = h \]

For our function \[ f(x) = 2(x - 3)^{2} + 1 \], the vertex is (3, 1). This vertex tells us the lowest point of the parabola since a > 0.

Vertical Stretch

The coefficient 'a' in the quadratic expression affects how 'stretched' or 'compressed' the parabola is vertically. When \[ |a| > 1 \], the parabola is vertically stretched, making it narrower. When \[ 0 < |a| < 1 \], the parabola is compressed, making it wider. For our function, \[ f(x) = 2(x - 3)^{2} + 1 \], the coefficient 2 indicates a vertical stretch. This means the parabola will be narrower compared to \[ f(x) = (x - 3)^{2} + 1 \]. This vertical stretch influences the curvature of the graph and directly impacts how steep or shallow the parabola appears.