How To Find The Y Intercept Of An Exponential Function

If the x-variable of a parent function, f(x), is replaced with 'x + two,' every point of the function will motion ii units left. Conversely, if the x-variable of a parent function, f(ten), is replaced with 'x - 2,' every point of the function will move 2 units correct.

The graph below demonstrates this for the office...

There are two important points to detect.

1. Point i: The asymptotes for the 3 functions are all the same. All of the asymptotes are y = 0 because horizontal shifts exercise not move horizontal lines. The asymptotes for exponential functions are always horizontal lines.
2. Indicate 2: The y-intercepts are unlike for the curves. Finding the location of a y-intercept for an exponential function requires a piffling work (shown below).

To determine the y-intercept of an exponential function, simply substitute nothing for the x-value in the function. The reason this works is because all points on the y-axis accept an x-value equal to zero.

Hither is the mathematics for all three of the functions that accept been graphed above. Colors have been added to match the graph in this section.

These y-intercepts can be verified by examining the graphs in this section.

The parent curve of an exponential function has the asymptote y = 0. This is evident by looking at the function...

...which has been graphed below.

Await what happens when we either add or decrease a number to/from our parent role.

Notice if we add the number i to the function that the part moves vertically up 1 unit of measurement. If we subtract 1 to the function, the function moves vertically down 1 unit. In both cases the asymptote follows the curve. The table below shows this close correlation.

If a part and its contrary are compared, we would see that they would be mirror versions of each other. The mirror would be the ten-axis, in this situation.

Nosotros should await at a specific situation. Allow us examine our parent office from a previous section and its opposite part.

Hither are their graphs.

This demonstrates how the transformed function is obtained past flipping the original function over the ten-axis.

Now that we have seen several types of transformations, we volition at present look at a few examples.

Case 1 : m(ten) = 4¹⁰⁺² + ane

Permit's kickoff decide how this function compares with its parent function, which is...

To graph g(x), we would have to movement h(ten) two units left and i unit up.

The number next to the x-value is the horizontal shift and we have to accept the opposite to decide the management of the shift. The +two really ways 2 units left. The +one is not adjacent to the x-value, which means it is the vertical shift number.

Now, we can sketch the graph of g(ten) since nosotros accept a full general idea of the shape of h(x), which is an exponential growth office.

The asymptote of h(x), which is y = 0, volition shift up i unit along with g(ten). This will make the asymptote of g(x) equal to y = 1.

To locate its y-intercept, we need to substitute the value 0 for the 10-value, like so.

Example two : thousand(x) = -two^ten-1 - iii

This transformation requires reflecting k(10) over the x-axis, moving the curve one unit right and 3 units down.

The asymptote must be y = -three, since the curve was moved downward 3 units.

The y-intercept can be found every bit follows.

Source: http://www.mathguide.com/lessons3/ExpFunctionsTrans.html

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