How are the graphs #f(x)=x# and #g(x)=3/4x^2# related?

Question

Elijah Allen · Accepted Answer

They both are expressions of a value derived from a given different value. They have common points at #x = 0 and x = 4/3#.

The first one is a straight line and the second one is a parabola. The have common points (intersections, solutions) at #x = 0 and x = 4/3#

Lincoln Anderson · Answer

undefined The graphs of $ f(x) = x $ and $ g(x) = \frac{3}{4}x^2 $ are related in the following ways:

1. Both functions are related by a transformation. The function $ g(x) $ is obtained from $ f(x) $ by squaring the input and then scaling the output by a factor of $ \frac{3}{4} $.
   
2. The graph of $ g(x) $ is a parabola that opens upwards, while the graph of $ f(x) $ is a straight line with a slope of 1, passing through the origin.

3. The vertex of the parabola $ g(x) $ is located at the origin (0,0), while the vertex of the linear function $ f(x) $ is also at the origin (0,0).

4. As $ x $ increases or decreases for both functions, $ f(x) = x $ produces a straight line with a constant slope of 1, while $ g(x) = \frac{3}{4}x^2 $ produces a parabola that opens upwards and widens as $ |x| $ increases.

5. Both functions are continuous and defined for all real numbers.

In summary, while $ f(x) = x $ represents a linear function, $ g(x) = \frac{3}{4}x^2 $ represents a quadratic function. Despite their different shapes, they share the commonality of passing through the origin (0,0).