Final answer:
The graphs of g(x) = −3x² - 5 and f(x) = x² are both quadratic functions, but with different coefficients. The graph of f(x) = x² is a standard upward-opening parabola that passes through the origin. The graph of g(x) = −3x² - 5 is also a downward-opening parabola, but with a steeper slope due to the negative coefficient of the quadratic term.
Step-by-step explanation:
When comparing the graphs of g(x) = −3x² - 5 and f(x) = x², we can see that they are both quadratic functions, but with different coefficients. The graph of f(x) = x² is a standard upward-opening parabola that passes through the origin. On the other hand, the graph of g(x) = −3x² - 5 is also a downward-opening parabola, but with a steeper slope due to the negative coefficient of the quadratic term.
For example, when we plug in x = 1, to f(x) = x², we get f(1) = 1² = 1. So, the point (1, 1) is on the graph of f(x) = x². However, when we plug in the same value for g(x) = −3x² - 5, we get g(1) = −3(1)² - 5 = -3 - 5 = -8. Hence, the point (1, -8) is on the graph of g(x) = −3x² - 5.
Overall, the graphs of g(x) = −3x² - 5 and f(x) = x² are similar in shape, both parabolas, but with different positioning and slopes.
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