The graphs of these quadratic functions are similar in shape, with the main differences being vertical shifts and y-intercepts. The graph of g(x)=5x^2 +3 is obtained by shifting the graph of f(x)=5x^2 −3 upward by 6 units.
The graphs of the quadratic functions f(x)=5x^2 −3 and g(x)=5x^2 +3
Both functions are quadratic, which means they have a graph in the shape of a parabola. The coefficient of the x^2 term in both functions is 5, indicating that the parabolas open upwards.
Now, let's analyze the differences:
Vertical Shift:
For f(x)=5x^2 −3, there is a vertical shift downward by 3 units due to the constant term -3.
For g(x)=5x^2 +3, there is a vertical shift upward by 3 units due to the constant term +3.
Y-Intercept:
The y-intercept of f(x) occurs when x=0, and f(0)=−3, so the y-intercept is (0, -3).
The y-intercept of g(x) occurs when x=0, and g(0)=3, so the y-intercept is (0, 3).
Overall Shape:
Both graphs have the same overall shape since the coefficient of the
x^2 term is the same in both functions.
Symmetry:
The parabolas are symmetric with respect to the y-axis, as changing
x to −x in the quadratic term does not affect the overall value of the function.