According to the given question information equation: f(x) =[tex]x^2[/tex]- 1 the solutions for f(x) are x = -1 and x = 1.
Difference from the axis of symmetry: For x = -1, the difference from the axis of symmetry x = 0 is -1 unit to the left. For x = 1, the difference from the axis of symmetry x = 0 is 1 unit to the right.
Equation: g(x) = [tex]x^2[/tex] + 3
since[tex]x^2[/tex] is always non-negative, there are no real solutions for this equation.
The graph of g(x) = [tex]x^2[/tex] + 3 does not intersect the x-axis.
Difference from the axis of symmetry: Since there are no real solutions for this equation, there are no differences from the axis of symmetry.
Equation: h(x) = [tex]x^2[/tex]
the solutions for h(x) are x = 0.
Difference from the axis of symmetry: For x = 0, the difference from the axis of symmetry x = 0 is 0 units.
In terms of the graphs, the graph of f(x) =[tex]x^2[/tex]- 1 will be symmetric with respect to the y-axis, while the graphs of[tex]g(x) = x^2 + 3 \\ and \\h(x) = x^2[/tex] will be symmetric with respect to the x-axis. The solutions x = -1 and x = 1 for f(x) will be 1 unit away from the axis of symmetry x = 0, on the left and right sides respectively.
What is equation?
An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=).
For example, the argument "2x + 3 = 9" asserts that the phrase "2x +3" equals the number "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true.
Let's find the solutions for each equation and identify their differences from the axis of symmetry.
Equation: f(x) =[tex]x^2[/tex]- 1
Axis of symmetry: The axis of symmetry for a parabola in the form [tex]y = ax^2 + bx + c[/tex]is given by x = -b/2a. In this case, a = 1 and b = 0, so the axis of symmetry is x = 0.
Solution: Setting f(x) = 0, we get[tex]x^2 - 1 = 0[/tex]. Solving for x, we have:
[tex]x^2 - 1 = 0\\x^2 = 1[/tex]
x = ±√1
So the solutions for f(x) are x = -1 and x = 1.
Difference from the axis of symmetry: For x = -1, the difference from the axis of symmetry x = 0 is -1 unit to the left. For x = 1, the difference from the axis of symmetry x = 0 is 1 unit to the right.
Equation: g(x) = [tex]x^2[/tex] + 3
Axis of symmetry: Similar to the previous equation, the axis of symmetry is x = 0.
Solution: Setting g(x) = 0, we get[tex]x^2 + 3 = 0.[/tex]However, since[tex]x^2[/tex] is always non-negative, there are no real solutions for this equation.
The graph of g(x) = [tex]x^2[/tex] + 3 does not intersect the x-axis.
Difference from the axis of symmetry: Since there are no real solutions for this equation, there are no differences from the axis of symmetry.
Equation: h(x) = [tex]x^2[/tex]
Axis of symmetry: Again, the axis of symmetry is x = 0.
Solution: Setting h(x) = 0, we get[tex]x^2[/tex] = 0. Solving for x, we have:
[tex]x^2[/tex]= 0
x = ±√0
So the solutions for h(x) are x = 0.
Difference from the axis of symmetry: For x = 0, the difference from the axis of symmetry x = 0 is 0 units.
In terms of the graphs, the graph of f(x) =[tex]x^2[/tex]- 1 will be symmetric with respect to the y-axis, while the graphs of[tex]g(x) = x^2 + 3 \\ and \\h(x) = x^2[/tex] will be symmetric with respect to the x-axis. The solutions x = -1 and x = 1 for f(x) will be 1 unit away from the axis of symmetry x = 0, on the left and right sides respectively.
The solutions x = 0 for g(x) and h(x) will be exactly on the axis of symmetry. So, on the graphs, the solutions will be at -1 and 1 units from the axis of symmetry for f(x), and at 0 units from the axis of symmetry for g(x) and h(x).
g(x) does not have any real solutions, as mentioned earlier, so it will not intersect the x-axis on the graph. The graph of h(x) will intersect the x-axis at x = 0, since x = 0 is a solution to the equation h(x) =[tex]x^2.[/tex]
Overall, the graph of f(x) will be shifted 1 unit to the right of the graph of h(x), with both being symmetric with respect to the y-axis, while the graph of g(x) will be symmetric with respect to the x-axis and not intersect the x-axis.
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