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Chapter 1: Problem 45

Exer. \(45-56:\) Sketch the graph of the equation. $$ y=2 x^{2}-1 $$

Short Answer

Expert verified
The graph is an upward-opening parabola with vertex (0, -1).

Step by step solution

01

Identify the Type of Equation

The given equation is a quadratic equation, which is a polynomial of degree 2. The general form of a quadratic equation is \( y = ax^2 + bx + c \).
02

Determine the Vertex

The equation \( y = 2x^2 - 1 \) is in standard form \(y = ax^2 + bx + c\) where \(a = 2\), \(b = 0\), and \(c = -1\). Since there's no linear term (\(b=0\)), the vertex of the parabola is at \(x = 0\). Substituting \(x = 0\) into the equation gives \(y = 2(0)^2 - 1 = -1\). Thus, the vertex is \((0, -1)\).
03

Determine the Direction of the Parabola

The coefficient \(a = 2\) is positive, which means the parabola opens upwards.
04

Identify the Axis of Symmetry

For a quadratic in the form \( y = ax^2 + bx + c \), the axis of symmetry is given by \( x = -\frac{b}{2a} \). Since \(b = 0\) in this equation, the axis of symmetry is \(x = 0\).
05

Calculate and Plot Additional Points

To sketch the graph, calculate additional points by substituting other values of \(x\) into the equation: - For \(x = 1\), \(y = 2(1)^2 - 1 = 1\).- For \(x = -1\), \(y = 2(-1)^2 - 1 = 1\).- For \(x = 2\), \(y = 2(2)^2 - 1 = 7\).- For \(x = -2\), \(y = 2(-2)^2 - 1 = 7\).Plot these points: \((1, 1)\), \((-1, 1)\), \((2, 7)\), and \((-2, 7)\).
06

Sketch the Graph

With the vertex \((0, -1)\), axis of symmetry \(x = 0\), and the points \((1, 1)\), \((-1, 1)\), \((2, 7)\), and \((-2, 7)\), sketch the parabola that opens upwards. The graph should be symmetrical around the axis.

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that you often encounter in mathematics, especially when dealing with quadratic equations. In a two-dimensional coordinate plane, the most common appearance of a parabola is the graph of a quadratic function. You can easily identify a parabola when you see an equation in the form of \( y = ax^2 + bx + c \). This equation outlines a curve where:
  • \( a \) determines how "wide" or "narrow" the parabola appears,
  • \( b \) affects the parabola's symmetry and position along the x-axis,
  • \( c \) moves the parabola up or down along the y-axis.
Parabolas can open upwards or downwards, and their shape makes them useful for modeling various real-world phenomena, from projectile motion to satellite dish shapes.
Vertex of a Parabola
The vertex of a parabola is a critical point where the curve changes direction. For an upward opening parabola like \( y = 2x^2 - 1 \), the vertex represents the lowest point on the graph. To find the vertex, you rely on the standard quadratic form \( y = ax^2 + bx + c \). In our particular equation, since \( b = 0 \), the formula simplifies:
  • The vertex is at \( x = -\frac{b}{2a} = 0 \),
  • Substituting \( x = 0 \) back into the equation, we compute \( y = 2(0)^2 - 1 = -1 \).
Thus, the vertex of this parabola is at \((0, -1)\). This point tells us that all other points on the graph are higher than or equal to this vertex, indicating the parabolic shape opening upwards.
Axis of Symmetry
The axis of symmetry is a vertical line that neatly divides the parabola into two mirror-image halves. For any quadratic function in the form of \( y = ax^2 + bx + c \), the axis of symmetry is invariably given by the equation \( x = -\frac{b}{2a} \). In the equation \( y = 2x^2 - 1 \):
  • \( b = 0 \), hence the axis of symmetry is \( x = 0 \).
This symmetry line runs through the vertex, effectively becoming the "spine" of the parabola. The importance of the axis of symmetry lies in its ability to simplify graphing and understanding the geometric properties of quadratic functions.
Upward Opening Graph
An upward opening graph describes a parabola that curves upwards, like the one in \( y = 2x^2 - 1 \). The direction the parabola opens is determined by the coefficient \( a \) in the quadratic form \( y = ax^2 + bx + c \). Because \( a = 2 \), which is positive:
  • The parabola opens upwards,
  • This means the vertex \((0, -1)\) is the minimum point.
Such parabolas are conducive to modeling situations where a minimum value is required, such as minimization problems in physics and engineering. On a graph, you sketch the parabola starting from the vertex and plotting additional points equally on both sides of the axis of symmetry, ensuring the curve smoothly ascends to create the "U" shape.

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