Photo AI
The graph of \( f(x) = 3^{-x} \) is sketched below - NSC Mathematics - Question 5 - 2020 - Paper 1
Question 5
The graph of \( f(x) = 3^{-x} \) is sketched below. A is the y-intercept of f. B is the point of intersection of f and the line \( y = 9 \). **5.1** Write down the ... show full transcript
Worked Solution & Example Answer:The graph of \( f(x) = 3^{-x} \) is sketched below - NSC Mathematics - Question 5 - 2020 - Paper 1
Step 1
Step 2
Determine the coordinates of B.
Answer
To find the coordinates of B, we need to solve for ( x ) when ( f(x) = 9 ).
Starting with the equation: [ 9 = 3^{-x} ] Converting 9 to the same base: [ 3^{2} = 3^{-x} ] Equating exponents gives us: [ -x = 2 ] From this, we find that ( x = -2 ). Hence, the coordinates of B are ( (-2, 9) ).
Step 3
Step 4
Describe the translation from f to \( h(x) = \frac{27}{3^{-x}} \).
Answer
The function ( h(x) = \frac{27}{3^{-x}} ) can be simplified as follows: [ h(x) = 27 \cdot 3^{x} ] This represents a vertical stretch of the original function ( f(x) ) and a shift to the right by 3 units. Thus the translation shifts the graph of f three units to the right.
Step 5
Determine the values of x for which \( h(x) < 1 \).
Answer
To find when ( h(x) < 1 ), we set up the inequality: [ 27 \cdot 3^{x} < 1 ] Dividing both sides by 27 gives: [ 3^{x} < \frac{1}{27} ] Since ( 27 = 3^{3} ), we have: [ 3^{x} < 3^{-3} ] Equating exponents leads to: [ x < -3 ] Thus, the values of ( x ) for which ( h(x) < 1 ) are ( x < -3 ).