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Chapter 2: Problem 41

In Exercises \(35-44,\) graph the quadratic function. $$f(x)=-2 x^{2}-12 x-16$$

Short Answer

Expert verified
Graph the function by identifying and plotting the vertex \((-3, 2)\), y-intercept \((0, -16)\), and points \((-2, 0)\) and \((-4, 0)\); the parabola opens downward.

Step by step solution

01

Identify the Form of the Quadratic

The quadratic function we are dealing with is in the standard form \(f(x) = ax^2 + bx + c\). Here, \(a = -2\), \(b = -12\), and \(c = -16\).
02

Find the Vertex

The vertex of a parabola described by a quadratic function \(ax^2 + bx + c\) can be found using the formula:\[ x = -\frac{b}{2a} \]. In this case, \(x = -\frac{-12}{2(-2)} = \frac{12}{-4} = -3\). Substitute \(x = -3\) back into the function to find \(f(-3)\):\[ f(-3) = -2(-3)^2 - 12(-3) - 16 = -18 + 36 - 16 = 2 \]. Thus, the vertex is \((-3, 2)\).
03

Determine if the Parabola Opens Upward or Downward

Since the coefficient \(a = -2\) is negative, the parabola opens downward.
04

Find the Y-intercept

The y-intercept occurs when \(x=0\). Substitute \(x=0\) into the function:\[ f(0) = -2(0)^2 - 12(0) - 16 = -16 \].So the y-intercept is \((0, -16)\).
05

Calculate Additional Points for Accuracy

To graph accurately, calculate additional points:When \(x = -2\):\[ f(-2) = -2(-2)^2 - 12(-2) - 16 = -8 + 24 - 16 = 0 \]Point: \((-2, 0)\).When \(x = -4\):\[ f(-4) = -2(-4)^2 - 12(-4) - 16 = -32 + 48 - 16 = 0 \]Point: \((-4, 0)\).
06

Draw the Parabola

Plot the vertex \((-3, 2)\), the y-intercept \((0, -16)\), and additional points \((-2, 0)\) and \((-4, 0)\) on a graph. Draw a smooth, symmetrical curve through these points, ensuring it opens downward.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Function
A quadratic function is a type of polynomial function that is represented in the general form:
  • \( f(x) = ax^2 + bx + c \)
This form consists of three main components:
  • The quadratic term: \( ax^2 \)
  • The linear term: \( bx \)
  • The constant term: \( c \)
The value of \(a\), \(b\), and \(c\) in the equation determine the shape and position of the parabolic graph.
The function in the original exercise, \( f(x) = -2x^2 - 12x - 16 \), is a perfect example of a quadratic function. The equation is characterized by the leading coefficient \( a = -2 \), which influences the direction in which the parabola opens. Quadratic functions graph as curves called parabolas, showing up in various scenarios in mathematics and practical applications.
Vertex of a Parabola
The vertex of a parabola is a crucial feature that indicates the maximum or minimum point of the curve. For any quadratic function in standard form, you can find the vertex using the formula:
  • \( x = -\frac{b}{2a} \)
This formula gives the x-coordinate of the vertex. To find the y-coordinate, substitute the x-value back into the quadratic equation.
In the given function \( f(x) = -2x^2 - 12x - 16 \), the vertex calculation goes as follows:
  • Calculate \( x = -\frac{-12}{2(-2)} = -3 \)
  • Substitute \( x = -3 \) back into the function to derive the y-coordinate: \( f(-3) = 2 \)
The vertex is therefore \( (-3, 2) \). This point helps in shaping the graph, determining the parabola's symmetry, and identifying the turning point as either a peak or trough.
Y-intercept
The y-intercept of a graph is the point where the parabola crosses the y-axis. It provides a starting reference when sketching a graph on a coordinate plane. You find this intercept by setting \( x = 0 \) in the quadratic equation and solving for \( f(x) \).
For our function \( f(x) = -2x^2 - 12x - 16 \), finding the y-intercept involves the following steps:
  • Substitute \( x = 0 \) into the function: \( f(0) = -16 \)
Thus, the y-intercept is the point \( (0, -16) \). This information helps locate the parabola within the coordinate system, providing essential insight into its orientation and vertical position.
Parabola Direction
The direction in which a parabola opens is determined by the sign and value of the coefficient \( a \) in the quadratic function \( ax^2 + bx + c \). Here are a few key points to remember:
  • If \( a > 0 \), the parabola opens upward.
  • If \( a < 0 \), the parabola opens downward.
In the case of the function \( f(x) = -2x^2 - 12x - 16 \), the value of \( a = -2 \) (which is negative) indicates that the parabola opens downward. Recognizing the direction is helpful when sketching the graph, as it affects the overall look and how the parabola behaves around its vertex.

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