Question

The functions f(x)=x²-1 and

g(x)=-x² +4 are shown on the graph.


Explain how to modify the graphs of f(x) and g(x) to graph the solution set to the following system of inequalities. How can the solution set be identified?
y>x²-1
ys-x²+4

Answer 1

The graph of f(x) should be reflected over the x-axis and then shifted up by 3 units in order to graph the solution set.

The solution set to this system of inequalities is the shaded region within the boundary lines.

In Mathematics, there are several rules that are generally used for writing and interpreting an inequality or system of inequalities that are plotted on a graph and these include the following:

• The line on a graph should be a dashed line when the inequality symbol is (> or <).
• The inequality symbol should be less than (<) when a dashed line is shaded below.
• The inequality symbol should be greater than or equal to (≥) when a solid line is shaded above.

Based on the graph shown above, the parent quadratic function
`f(x)=x^(2) -1` was reflected over the x-axis and shifted up by 3 units in order to produce the transformed function
`g(x) =-x^(2) +4`;

g(x) = -f(x) + 3

`g(x)=-(x^(2) -1)+3\\\\g(x) =-x^(2) +4`

In this context, we can logically deduce that the graph of this quadratic inequality,
`y > x^2-1` would be a dashed line and shaded above while the graph of this quadratic inequality,
`y\leq x^(2) +4` would be a dashed line and shaded below.

In conclusion, the solution set is the region shaded purple on the graph within the boundary lines.