Asosiy tarkibga o'tish

6.1-bo'lim : Eksponensial funksiyalar

3. Quyidagilarning har birini eskiz qiling.

  1. \(f\left( x \right) = {6^x}\)
  2. \(g\left( x \right) = {6^x} - 9\)
  3. \(g\left( x \right) = {6^{x + 1}}\)
a Yechim

Biz ushbu funktsiyaning grafigi uchun chizishimiz mumkin bo'lgan tezkor qiymatlar jadvalini yaratishimiz mumkin.

\(x\) \(f\left( x \right)\)
-2 \(f\left( { - 2} \right) = {6^{ - 2}} = \frac{1}{{{6^{\,2}}}} = \frac{1}{{36}}\)
-1 \(f\left( { - 1} \right) = {6^{ - 1}} = \frac{1}{6}\)
0 \(f\left( 0 \right) = {6^0} = 1\)
1 \(f\left( 1 \right) = {6^1} = 6\)
2 \(f\left( 2 \right) = {6^2} = 36\)

Mana, funksiya grafigining qisqacha eskizi.

The graph of \(f\left( x \right) = {6^x}\) on \( - 2 \le x \le 2\). The graph is always positive, is nearly flat along the x-axis for negative \(x\) and increases very steeply to the right. The points that were computed in the table above are marked on the graph and include \(\left( {0,1} \right)\), \(\left( {1,6} \right)\) and \(\left( {2,36} \right)\).

b Yechim

Ushbu qism uchun biz qilishimiz kerak bo'lgan narsa - eslash Transformatsiyalar bir necha bobdan oldingi qism. \(f\left( x \right) = {6^x}\) ning “asosiy” funksiyasidan foydalanib, ushbu qism uchun funksiya quyidagicha yozilishi mumkin:

\[g\left( x \right) = {6^x} - 9 = f\left( x \right) - 9\]

Shuning uchun, bu qism uchun grafik faqat 9 ga pastga siljigan \(f\left( x \right)\) grafigi.

Ushbu funktsiyaning grafigi quyida ko'rsatilgan. Moviy chiziqli chiziq “tayanch” funksiyasi, \(f\left( x \right)\) va qizil qattiq chiziq bu qismning grafigi, \(g\left( x \right)\).

The graph of \(g\left( x \right) = {6^x} - 9\). The blue dashed line is the “base” function \(f\left( x \right) = {6^x}\) and the solid red line is \(g\left( x \right)\), which is exactly the same shape shifted down by 9. The solid graph flattens out along the horizontal line \(y = - 9\) on the left instead of along the x-axis and crosses the x-axis just to the right of \(x = 1\).

c Yechim

Ushbu qism uchun biz qilishimiz kerak bo'lgan narsa - eslash Transformatsiyalar bir necha bobdan oldingi qism. \(f\left( x \right) = {6^x}\) ning “asosiy” funksiyasidan foydalanib, ushbu qism uchun funksiya quyidagicha yozilishi mumkin:

\[g\left( x \right) = {6^{x + 1}} = f\left( {x + 1} \right)\]

Shuning uchun, bu qism uchun grafik faqat 1 ga chapga siljigan \(f\left( x \right)\) grafigi.

Ushbu funktsiyaning grafigi quyida ko'rsatilgan. Moviy chiziqli chiziq “tayanch” funksiyasi, \(f\left( x \right)\), qizil qattiq chiziq esa bu qismning grafigi, \(g\left( x \right)\).

The graph of \(g\left( x \right) = {6^{x + 1}}\). The blue dashed line is the “base” function \(f\left( x \right) = {6^x}\) and the solid red line is \(g\left( x \right)\), which is exactly the same shape shifted left by 1. Both graphs flatten out along the x-axis on the left and increase steeply on the right, with the solid graph always one unit to the left of the dashed one.