Given the function f(x) = 3^{2-x} - 3, 6.1 Write f in the form f(x) = ab^x + q. 6.2 Hence, or otherwise, sketch the graph of f showing the asymptote and the intercepts with the axes.
Answer (1)
To write the function f ( x ) = 3 2 − x − 3 in the form f ( x ) = a b x + q , we need to break this down step-by-step.
Step 1: Rewrite the Expression
Start from the original function: f ( x ) = 3 2 − x − 3 .
Notice that 3 2 − x can be rewritten using the property of exponents as 3 2 ⋅ 3 − x .
Step 2: Identify a , b , and q
Rewriting the expression, f ( x ) = ( 3 2 ⋅ 3 − x ) − 3 = 9 ⋅ 3 − x − 3 .
In this form, a = 9 , b = 3 1 due to the negative exponent, and q = − 3 .
Step 3: Hence, Sketch the Graph
Asymptote:
As x → ∞ , 3 − x → 0 , so f ( x ) → − 3 . This means the horizontal asymptote is y = − 3 .
Intercepts:
Y-intercept: Set x = 0 : f ( 0 ) = 9 ⋅ 3 0 − 3 = 9 ⋅ 1 − 3 = 6 So, the y-intercept is at ( 0 , 6 ) .
X-intercept: Solve f ( x ) = 0 : 9 ⋅ 3 − x − 3 = 0 9 ⋅ 3 − x = 3 3 − x = 3 1 Convert it back to exponential form: − x = − 1 ⇒ x = 1 Thus, the x-intercept is at ( 1 , 0 ) .
Sketching the Graph:
Draw the horizontal line y = − 3 for the asymptote.
Mark the points for the y-intercept ( 0 , 6 ) and the x-intercept ( 1 , 0 ) .
Since the base of the exponent is 3 1 , the function is decreasing as x increases.
By following these steps, you have a function expressed in the form f ( x ) = a b x + q , with visual features like asymptotes and intercepts identified for graphing.
