The graph below shows two functions: function f of x is a straight line that joins the ordered pairs negative 2.5, 9, and 2.5, negative 1. Function g of x is a curved line that joins the ordered pairs negative 1.4, 6.8, and 2, 0.2 Based on the graph, what are the approximate solutions to the equation -2x + 4 = (0.25)x?
Solution:
The above diagram is presented again with two points marked as A and B on the graph.
Point A is (2,0) and it happens to be the intersection point of the straight line f(x) - 2x+4 and the curve \(0.25^{x}\).
The other point of intersection is B which is approximately which is approximately (-1.4, 6.8). Since these points are points of intersection which implies that these satisfy both the equations i.e. f(x) = -2x + 4 and g(x) = \(0.25^{x}\).
Now let us verify that these points satisfy these two equations. This can be done by substituting these coordinates into the respective equations and see whether they give the same result.
Point A - (2,0)
f(x) = -2(2) + 4 = 0
g(x) = \(0.25^{x}\)= 0.25² = 0.0625 which close to zero. Since \(0.25^{x}\) is an exponential function i.e. something raised to the power of something, it will never be completely zero, it will tend towards zero but never reach there.
Point B - (-1.4, 6.8)
f(x) = -2(-1.4) + 4 = 6.8
g(x) = \(0.25^{1.4}\) = 0.25-1.4 = 6.96 which is close to 6.8
Hence the approximate values of the solution to the equation -2x + 4 = \(0.25^{x}\) are x = 2 and x = -1.4
The graph below shows two functions: function f of x is a straight line which joins the ordered pairs negative 2.5, 9 and 2.5, negative 1. Function g of x is a curved line which joins the ordered pairs negative 1.4, 6.8 and 2, 0.2 Based on the graph, what are the approximate solutions to the equation -2x + 4 = (0.25)x?
Summary:
Based on the graph, what are the approximate solutions to the equation −2x + 4 = \(0.25^{1.4}\) are x = 2 and x = -1.4