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Chapter 0: Problem 32

Graph each function. $$g(x)=-x^{2}+1$$

Short Answer

Expert verified
Plot the vertex (0,1) and additional points (1,0), (-1,0), (2,-3), (-2,-3), then draw a downward-opening parabola.

Step by step solution

01

Identify the type of function

The function given is a quadratic function of the form \(g(x) = ax^2 + bx + c\). Here, \(a = -1\), \(b = 0\), and \(c = 1\). Quadratic functions graph as parabolas.
02

Determine the direction of the parabola

The coefficient \(a = -1\) is negative, indicating that the parabola opens downwards.
03

Find the vertex of the parabola

The vertex form of a quadratic function is \(g(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex. For the given equation \(g(x) = -x^2 + 1\), the vertex is at \((0, 1)\).
04

Find additional points on the graph

Choose a few values of x to find additional points. For example: \[ g(1) = -(1)^2 + 1 = 0, \ g(-1) = -(-1)^2 + 1 = 0, \ g(2) = -(2)^2 + 1 = -3, \ g(-2) = -(-2)^2 + 1 = -3 \] This gives the points \( (1, 0), \ (-1, 0), \ (2, -3) \) and \((-2, -3)\).
05

Draw the graph

Plot the vertex \((0,1)\) and the additional points \( (1, 0), (-1, 0), (2, -3), (-2, -3) \). Draw a smooth curve through these points to form the parabola opening downwards.

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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 can be written in the form \(ax^2 + bx + c\). It is called 'quadratic' because the highest power of the variable 'x' is squared.
In the case of our function \(g(x) = -x^2 + 1\), the coefficients are \(a = -1\), \(b = 0\), and \(c = 1\).
Quadratic functions create unique graphs that are always parabolas.
Understanding these functions is key in algebra and helps in various real-world applications, such as physics and engineering.
parabola
A parabola is the graph of a quadratic function.
It has a distinct U-shaped curve.
Depending on the coefficient 'a' in front of the \(x^2\) term, the parabola can open upwards or downwards.
In our example, the function \(g(x) = -x^2 + 1\) creates a parabola that opens downward because the coefficient \(a\) is negative.
Parabolas have specific properties, like symmetry around a central axis, making them easily recognizable and useful in mathematical modeling.
vertex
The vertex of a parabola is its highest or lowest point, depending on the direction it opens.
For \(g(x) = -x^2 + 1\), the vertex form of a quadratic function \(g(x) = a(x-h)^2 + k\) helps us identify the vertex.
In this equation, 'h' and 'k' are the x and y coordinates of the vertex, respectively.
Here, the vertex is at \(h = 0\), and \(k = 1\), giving us the point \( (0, 1)\).
This is the peak point because the parabola opens downward.
direction of parabola
The direction in which a parabola opens is determined by the sign of the coefficient 'a' in the quadratic function.
If \(a > 0\), the parabola opens upwards, like a U-shape. Conversely, if \(a < 0\), it opens downwards, like an upside-down U.
In our case, \(a = -1\), indicating that our parabola opens downward.
It's essential to understand this because it affects the positioning of the vertex and the general shape of the graph.
additional points
To graph the quadratic function accurately, finding additional points around the vertex is very helpful.
These points allow us to shape the parabola correctly.
For \(g(x) = -x^2 + 1\), we substitute several values of 'x' to find corresponding 'y' values.
For example:
  • \(g(1) = -1^2 + 1 = 0\)
  • \(g(-1) = -(-1)^2 + 1 = 0\)
  • \(g(2) = -2^2 + 1 = -3\)
  • \(g(-2) = -(-2)^2 + 1 = -3\)
These points \( (1, 0)\), \( (-1, 0)\), \( (2, -3)\), and \( (-2, -3)\) helps us complete the graph.

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