Question

Complete the square of each quadratic expression. Then graph each function using graphing techniques. $$ f(x)=-2 x^{2}-12 x-13 $$

Short answer

Complete the square to get \( f(x) = -2(x + 3)^{2} + 5 \). The vertex is at \((-3, 5)\), and the parabola opens downward.

Step-by-step solution

Write the Quadratic in Standard Form

Given the quadratic function: \[ f(x) = -2x^{2} - 12x - 13 \] Verify it is in standard form, which is \( ax^2 + bx + c \).

Factor out the Leading Coefficient

Factor out \(-2\) from the first two terms to set up completing the square: \[ f(x) = -2(x^{2} + 6x) - 13 \]

Complete the Square

To complete the square inside the parentheses, take half the coefficient of \(x\), which is 6, and square it: \[ (6 / 2)^2 = 9 \] Add and subtract this value inside the parentheses: \[ f(x) = -2(x^{2} + 6x + 9 - 9) - 13 \] Simplify the expression inside the parentheses: \[ f(x) = -2((x + 3)^{2} - 9) - 13 \]

Distribute and Simplify

Distribute the \(-2\) and then simplify: \[ f(x) = -2(x + 3)^{2} + 18 - 13 \] Combine the constants: \[ f(x) = -2(x + 3)^{2} + 5 \]

Graphing the Function

The function is now in vertex form \( a(x-h)^{2} + k \), where the vertex \( (h, k) \) is \( (-3, 5) \). The direction of the parabola is downwards because the coefficient of \( (x + 3)^{2} \) is negative. Plot the vertex \((-3, 5)\) and sketch the parabola opening downwards.

Key concepts

Quadratic Function

A quadratic function is a second-degree polynomial of the form \[ f(x) = ax^{2} + bx + c \]. The graph of a quadratic function is a parabola, which can open upwards or downwards. The direction of the parabola is determined by the sign of the coefficient \(a\).

If \(a\) is positive, the parabola opens upwards, creating a U-shape. If \(a\) is negative, the parabola opens downwards, forming an inverted U-shape.

The quadratic function given in the exercise is \[ f(x) = -2x^{2} - 12x - 13 \], where \(a = -2\), \(b = -12\), and \(c = -13\). Because \(a\) is negative, the parabola will open downwards. Understanding the general shape and direction of the quadratic function helps us better visualize and graph the equation.

Graphing Techniques

Graphing a quadratic function involves several key steps to ensure accuracy and clarity. Here are the main techniques:

• Identify the Vertex: Use the vertex form \[ f(x) = a(x-h)^{2} + k \]. Calculate the vertex (h, k). In the solved exercise, the vertex was \[ (-3,5) \]. Plot this point first.
• Determine the Direction: Check the sign of \(a\). If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
• Find Additional Points: Choose x-values around the vertex and calculate corresponding y-values to get more points for the graph.
• Axis of Symmetry: The line \(x = h \) represents the axis of symmetry, where the parabola is mirrored.
• Plot the Parabola: Connect the points smoothly to draw the shape of the parabola, making sure it extends in the correct direction.

Utilizing these techniques systematically helps in accurately plotting a quadratic function.

Vertex Form

The vertex form of a quadratic function is a useful format for graphing and understanding the properties of the parabola. The vertex form is given by \[ f(x) = a(x-h)^{2} + k \], where \( (h, k) \) is the vertex.

• Finding the Vertex: The vertex \( (h, k) \) is the highest or lowest point of the parabola depending on the sign of \(a\). For our function \[ f(x) = -2(x + 3)^{2} + 5 \], the vertex is \( (-3, 5) \). This represents the maximum point because the parabola opens downwards.
• Advantages of Vertex Form: It directly shows the vertex, making graphing easier. Also, transformations like shifts and stretches can be recognized immediately.
• Converting to Vertex Form: Completing the square is a common method to convert a quadratic function from standard form \[ f(x) = ax^{2} + bx + c \] to vertex form. Follow steps to complete the square, and factor the quadratic expression so it fits the vertex form.

Understanding and using the vertex form allows for efficient graphing and analysis of the quadratic function.