document.write( "Question 391614: write an exponential function whose graph passes through the points (0,4) and (-2,100) \n" ); document.write( "
Algebra.Com's Answer #277829 by jsmallt9(3758)\"\" \"About 
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The general form for an exponential function is:
\n" ); document.write( "\"f%28x%29+=+a%2Ab%5Ex\"
\n" ); document.write( "where \"a\" and \"b\" are non-zero constants. So the task before us is to figure out the specific \"a\" and \"b\" that will cause the graph to pass through the two given points.

\n" ); document.write( "If the graph passes through (0, 4) then when we use an input of 0 for the function we should get 4 as the output. IOW: f(0) = 4. Substituting 0 and 4 into the general equation we get:
\n" ); document.write( "\"4+=+a%2Ab%5E0\"
\n" ); document.write( "Since \"b%5E0\" equals 1 this becomes:
\n" ); document.write( "4 = a*1
\n" ); document.write( "or
\n" ); document.write( "4 = a
\n" ); document.write( "Already we have our \"a\".

\n" ); document.write( "If the graph passes through (-2, 100) then when we use an input of -2 for the function we should get 100 as the output. IOW: f(-2) = 100. Substituting -2 and 100 (and our \"a\" (4) into the general equation we get:
\n" ); document.write( "\"100+=+%284%29%2Ab%5E%28-2%29\"
\n" ); document.write( "Dividing both sides by 4 we get:
\n" ); document.write( "\"25+=+b%5E%28-2%29\"
\n" ); document.write( "Next we need to find a way to change the exponent on \"b\" to a 1. The way to do this is to raise both sides of the equation to the reciprocal of -2 power. The reasons we do this are:
  • Raising a power to a power means we will multiply the exponents.
  • When multiplying reciprocals the answer is always a 1!

\n" ); document.write( "The reciprocal of -2 is -1/2. So now we have:
\n" ); document.write( "\"%2825%29%5E%28%28-1%2F2%29%29+=+%28b%5E%28-2%29%29%5E%28%28-1%2F2%29%29\"
\n" ); document.write( "On the right side we get \"b%5E1\" or \"b\" just as we planned. All we have to do is simplify the left side. If you have trouble with negative and/or fraction exponents I find it helpful to factor the exponent in a certain way:
  • If the exponent is negative, factor out a -1.
  • If the exponent is fractional and the numerator is not a 1, factor out the numerator, (For example, factor an exponent like \"3%2F4\" into \"3%2A%281%2F4%29\").

\n" ); document.write( "Factoring our exponent this way we get:
\n" ); document.write( "\"%2825%29%5E%28%28%28-1%29%2A%281%2F2%29%29%29+=+b\"
\n" ); document.write( "Looking at the factors of the exponent we see a -1 and a 1/2. The -1 (as an exponent) tells us that a reciprocal will be found. And the 1/2 (as an exponent) means a square root will be found. And since multiplication is Commutative, we can do these operations in any order we choose! Since finding a square root of 25 seems easier than the reciprocal, I choose to start with that. Then we will finish with a reciprocal:
\n" ); document.write( "\"%2825%29%5E%28%28%281%2F2%29%2A%28-1%29%29%29+=+b\"
\n" ); document.write( "\"%2825%5E%28%281%2F2%29%29%29%5E%28-1%29+=+b\"
\n" ); document.write( "\"%285%29%5E%28-1%29+=+b\"
\n" ); document.write( "\"1%2F5+=+b\"

\n" ); document.write( "Now that we have our \"a\" and \"b\" we can write the desired function:
\n" ); document.write( "\"f%28x%29+=+4%2A%281%2F5%29%5Ex\"
\n" ); document.write( "or (if you prefer decimals):
\n" ); document.write( "\"f%28x%29+=+4%2A%280.2%29%5Ex\" \n" ); document.write( "
\n" );