Question

Complete the square of each quadratic expression. Then graph each function using graphing techniques. $$ f(x)=-3 x^{2}-12 x-17 $$

Short answer

The function in vertex form is \(f(x) = -3(x + 2)^2 - 5\). The vertex is at \((-2, -5)\) and the parabola opens downwards.

Step-by-step solution

Write the quadratic expression in standard form

The given quadratic expression is already in standard form: \[f(x) = -3x^2 - 12x - 17\]

Factor out the leading coefficient from the first two terms

Factor out the \(-3\) from the terms with \(x^2\) and \(x\)\[-3(x^2 + 4x) - 17\]

Complete the square inside the parentheses

To complete the square, take half of the \(4\) (coefficient of \(x\)), square it, and add and subtract it inside the parentheses.\[-3(x^2 + 4x + 4 - 4) - 17\]Simplify:\[-3((x + 2)^2 - 4) - 17\]

Distribute the leading coefficient

Distribute \(-3\) through the terms inside the parentheses.\[-3(x + 2)^2 + 12 - 17\]Simplify:\[-3(x + 2)^2 - 5\]

Write the function in vertex form

The function in vertex form is:\[f(x) = -3(x + 2)^2 - 5\]This shows that the vertex of the parabola is at \((-2, -5)\)

Graph the function using the vertex and direction

Since the coefficient of \((x + 2)^2\) is negative, the parabola opens downwards. The vertex \((-2, -5)\) is the maximum point.Use this information to plot the vertex and sketch the parabola.

Key concepts

Quadratic Functions

A quadratic function is a type of polynomial function with a degree of 2. This means the highest exponent of the variable (usually denoted as 'x') is 2. The general form of a quadratic function is \[f(x) = ax^2 + bx + c\].
Here, 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These functions graph as parabolas, which are U-shaped curves.
Let's consider the given quadratic function from the exercise: \[f(x) = -3x^2 - 12x - 17\].

When working with quadratic functions, you often need to rewrite the equation in different forms to understand its properties better, such as the vertex form when graphing.

Vertex Form

The vertex form of a quadratic function provides a clear way to identify the vertex of the parabola, which is either the highest or lowest point of the graph. The vertex form is written as \[f(x) = a(x - h)^2 + k\], where \(h, k\) is the vertex of the parabola.

To convert our given function into vertex form, we use the method of completing the square. Here are the steps:

• Factor out the leading coefficient from the terms with \x^2\ and \x\: \f(x) = -3(x^2 + 4x) - 17\
• Complete the square inside the parentheses: \[f(x) = -3(x^2 + 4x + 4 - 4) - 17\]
  Simplify: \[f(x) = -3((x + 2)^2 - 4) - 17\]
• Distribute the \-3\: \[f(x) = -3(x + 2)^2 + 12 - 17\]
• Simplify further: \[f(x) = -3(x + 2)^2 - 5\]

Now, we can easily see that the vertex of the parabola is \(-2, -5\). This makes graphing the function much simpler, as we know the peak point.

Graphing Techniques

Once you have the quadratic function in vertex form, graphing becomes straightforward. Here are the steps to graph \[f(x) = -3(x + 2)^2 - 5\]:

• Identify the vertex \(-2, -5\): This is the highest point because the parabola opens downwards (since the coefficient of the squared term is negative).
• Determine the direction: The coefficient of \((x + 2)^2\) is -3, which means the parabola opens downwards.
• Plot the vertex on the graph: Place a point at \(-2, -5\).
• Find additional points: Choose x-values around the vertex to help sketch the curve. For example, if x = -1, f(x) = -3(-1 + 2)^2 - 5 = -8. So the point is (-1, -8). If x = -3, f(x) = -3(-3 + 2)^2 - 5 = -8. So the point is (-3, -8). Plot these points.
• Draw the parabola: Connect the points in a smooth, U-shaped curve.

By following these simple steps, you can accurately graph any quadratic function.