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Chapter 1: Problem 31

Graph each equation by plotting points that satisfy the equation. $$y=x^{2}-3$$

Short Answer

Expert verified
Several points are plotted on a graph based on selected x values and their corresponding y values as determined by the equation \(y = x^2 - 3\). When these points are connected, they form a parabola that opens downward.

Step by step solution

01

Setup

Select several values for x to be substituted into the equation. It is helpful to select a range of positive and negative values, as well as zero.
02

Calculate y values

Substitute the chosen x values into the equation \(y = x^2 - 3\) to obtain the corresponding y values. The equation is applied by squaring each x value and subtracting 3 from the result.
03

Create coordinate pairs

Pair each x value with its corresponding y value to form coordinate pairs (x, y). These pairs represent points that satisfy the equation and will be plotted on the graph.
04

Plot the points

Plot the points on a graph, remembering that each coordinate pair (x, y) represents a point where x is the horizontal coordinate and y is the vertical coordinate.
05

Draw the curve

Join the plotted points to form a continuous curve that represents the graph of the equation \(y= x^2-3\). The curve should be a downward-opening parabola, given the negative constant in the equation.

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Key Concepts

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

Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In our exercise, the specific quadratic equation is \(y = x^2 - 3\). This equation features no linear term (the \(b\) value is zero) and a constant of \(-3\). The quadratic term, \(x^2\), means that the graph will always be a parabola. Quadratics are known for shaping into U-like curves either upwards or downwards, dependent on the sign of the leading coefficient \(a\). For our equation, since \(a = 1\), the parabola opens upwards.
Coordinate Pairs
Coordinate pairs are essential in graphing functions. Each coordinate pair is structured as \((x, y)\), capturing the point on the graph where a plotted x-value yields a y-value through a given function. For graphing \(y = x^2 - 3\), you'll begin by choosing several x-values.
  • Select values such as \(-2, -1, 0, 1,\) and \(2\).
  • Then plug these x-values into the equation to find corresponding y-values.
  • For instance, if \(x = 0\), the calculation leads to \(y = 0^2 - 3 = -3\). So the coordinate pair is \((0, -3)\).
These pairs will be the specific points you'll mark on the graph.
Parabola
A parabola is a symmetrical, U-shaped curve depicted usually when graphing quadratic functions. The parabola for our equation \(y = x^2 - 3\) will open upwards as its \(x^2\) term has a positive coefficient of 1. Here鈥檚 what to remember about this parabola:
  • The vertex of the parabola represents the minimum point which in this case happens at the coordinate \((0, -3)\).
  • Parabolas are symmetrical about a vertical line passing through the vertex, known as the axis of symmetry. For \(y = x^2 - 3\), this axis is \(x = 0\).
  • As you move away from the vertex along the x-axis, the y-values will increase symmetrically.
By understanding these characteristics, you can accurately predict and draw the shape of the parabola on a graph.
Plotting Points
Plotting points is a straightforward method for graphing equations. Start with coordinate pairs that you derived from substituting x-values into the equation \(y = x^2 - 3\). Here's how you can do this:
  • Begin by drawing your axes. The x-axis is horizontal and the y-axis is vertical.
  • Next, mark the points on the graph paper using the coordinate pairs you have, such as \((-2, 1)\), \((-1, -2)\), \((0, -3)\), \((1, -2)\), and \((2, 1)\).
  • Once all points are plotted, smoothly connect them with a curve.
This continuous curve represents the quadratic function. It shows how y changes with varying x-values, embodying the characteristics of a parabola in visual form.

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