Write an equation in the form 𝑦 equals 𝑏 to the power of 𝑥 for the numbers in the table.
We are given four sets of values. When 𝑥 equals zero, 𝑦 is equal to one; when 𝑥 equals one, 𝑦 is negative two; when 𝑥 equals two, 𝑦 is equal to four; and when 𝑥 is equal to three, 𝑦 is equal to negative eight. In order to work out the value of the constant 𝑏, we need to substitute our values of 𝑥 and 𝑦 into the equation. Substituting 𝑥 equals zero and 𝑦 equals one into the equation gives us one equals 𝑏 to the power of zero. As any value to the power of zero is equal to one, this does not help us calculate the value of 𝑏.
Substituting 𝑥 equals one and 𝑦 equals negative two gives us negative two is equal to 𝑏 to the first power or 𝑏 to the power of one. As anything to the power of one is equal to itself, 𝑏 is equal to negative two. Substituting this back into our equation gives us 𝑦 is equal to negative two to the power of 𝑥. We can check this answer is correct by substituting the third and fourth pairs of values.
When 𝑥 is equal to two and 𝑦 is equal to four, we have four is equal to negative two squared. Squaring a number means multiplying it by itself, and multiplying a negative by a negative gives a positive answer. Therefore, negative two squared is equal to four. Substituting our final pair of values gives us negative eight is equal to negative two cubed. This is the same as multiplying negative two by negative two and then negative two again. Four multiplied by negative two is negative eight. Therefore, negative two cubed is also negative eight. This confirms that the correct equation is 𝑦 is equal to negative two to the power of 𝑥.