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Chapter 3: Problem 14

Graph the given functions. $$y=-2 x^{2}$$

Short Answer

Expert verified
The graph of \(y = -2x^2\) is a downward-opening parabola with vertex at (0,0).

Step by step solution

01

Understanding the Given Function

The given quadratic function is \(y = -2x^2\). This is a polynomial of degree 2, with the coefficients indicating a parabolic graph.
02

Determine the Vertex

Since the function is in the standard form \(y = ax^2 + bx + c\), and there are no \(x\)-term or constant term, the vertex point of the parabola is \((0, 0)\).
03

Identifying the Direction of the Parabola

The coefficient \(-2\) in front of \(x^2\) is negative, which means the parabola opens downwards.
04

Plotting Key Points

To better visualize the parabola, plot several points around the vertex. For \(x = 1\), \(y = -2(1)^2 = -2\); for \(x = -1\), \(y = -2(-1)^2 = -2\). Similarly, for \(x = 2\), \(y = -2(2)^2 = -8\); for \(x = -2\), \(y = -2(-2)^2 = -8\). This gives points \((1, -2)\), \((-1, -2)\), \((2, -8)\), and \((-2, -8)\).
05

Sketch the Graph

Using the vertex \((0, 0)\) and the points calculated, plot them on a coordinate plane. Connect these points with a smooth curve to represent the parabola opening downward through the vertex.

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Key Concepts

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

Vertex Identification
In a quadratic function like \(y = ax^2 + bx + c\), the vertex acts as an anchor point of the parabola. Since the given equation, \(y = -2x^2\), does not contain an \(x\)-term (\
Parabola Direction
The direction of a parabola is determined by the sign of the coefficient \(a\) in the quadratic equation \(y = ax^2 + bx + c\). For our equation, \(y = -2x^2\), \(a = -2\).

  • A positive \(a\) value would mean the parabola opens upwards, as the arms of the parabola reach towards positive infinity.
  • A negative \(a\) value, as we have here, indicates the parabola opens downwards. This means the arms of the parabola extend towards negative infinity.
Understanding this is crucial when sketching or interpreting a parabola on a graph.
It impacts the overall shape and direction of the curve, showing how the graph behaves as \(x\) moves further from the vertex.
Plotting Points
Plotting points is a practical step to ensure accurate graphing of quadratic functions. Here, we start by calculating values of \(y\) for various \(x\) values around the vertex, providing reference points to draw the curve accurately.
  • For \(x = 0\), we already know the vertex is \((0, 0)\).
  • Choose simple values such as \(x = 1\) and \(x = -1\). Substituting these into the equation gives us \((1, -2)\) and \((-1, -2)\).
  • Increase the span, testing values further from the vertex, like \(x = 2\) and \(x = -2\), resulting in \((2, -8)\) and \((-2, -8)\).
By plotting these points on a coordinate plane, they offer a visual guideline to connect and shape the parabola. Using these key points, one can sketch a smooth curve that passes through them all, accurately depicting the parabola's downturn.

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Most popular questions from this chapter

In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of \(y=x\) and \(y=|x|\) on the same coordinate system. Explain why the graphs differ.

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