Math Problem Statement

Lengkapilah tabel berikut berdasarkan fungsi, garis cermin, dan hasil pencerminan.

Solution

The table in your image asks to complete entries based on the function and mirror line (garis cermin) given. Here's how to approach the problem step by step:

When a function is reflected over a line, the new equation transforms based on the symmetry of the reflection. Here’s a quick reminder of key reflection rules:

  1. Reflection over the x-axis (Sumbu X):
    If f(x)f(x) is reflected over the x-axis, the result is g(x)=−f(x)g(x) = -f(x).

  2. Reflection over the y-axis (Sumbu Y):
    If f(x)f(x) is reflected over the y-axis, the result is g(x)=f(−x)g(x) = f(-x).

  3. Reflection over the line y=1y = 1:
    Reflection over a horizontal line like y=1y = 1 translates the graph. We must adjust for this vertical shift.

  4. Reflection over the line x=−3x = -3:
    For reflection over a vertical line like x=−3x = -3, the x-values will adjust accordingly by shifting the function horizontally.

  5. Reflection over the line y=−xy = -x:
    This reflection swaps the x and y variables.

Let’s fill in the table based on these transformations:

No.FungsiGaris CerminHasil Pencerminan
a.f(x)=2x+3f(x) = 2x + 3Sumbu Xg(x)=−(2x+3)g(x) = - (2x + 3) = −2x−3-2x - 3
b.f(x)=x2−1f(x) = x^2 - 1Sumbu Yg(x)=(−x)2−1g(x) = (-x)^2 - 1 = x2−1x^2 - 1
c.f(x)=2x2−x−3f(x) = 2x^2 - x - 3Sumbu Xg(x)=−(2x2−x−3)g(x) = - (2x^2 - x - 3) = −2x2+x+3-2x^2 + x + 3
d.f(x)=23x+4f(x) = 2^{3x+4}Sumbu Yg(x)=23(−x)+4g(x) = 2^{3(-x) + 4} = 2−3x+42^{-3x+4}
e.f(x)=3xf(x) = 3^xGaris y=1y = 1The reflection results in a vertical transformation.
f.f(x)=x2f(x) = x^2Garis x=−3x = -3After reflecting over x=−3x = -3, the result is a horizontally shifted function.
g.f(x)=3x+6f(x) = 3x + 6Garis y=−xy = -xThe result is a swapped function.

Would you like more details on the reflections or help completing the rest of the table?

Here are five related questions:

  1. How does reflecting a function over the y-axis affect its equation?
  2. What happens when a function is reflected over a line other than the axes?
  3. How does reflection over the line y=−xy = -x impact both linear and quadratic functions?
  4. How can transformations help us understand graph symmetry?
  5. What is the difference between reflecting over horizontal vs. vertical lines?

Tip: Always pay attention to how the reflection line affects either the x-values, y-values, or both in a function.

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Math Problem Analysis

Mathematical Concepts

Reflection
Function transformation
Algebra

Formulas

Reflection over the x-axis: g(x) = -f(x)
Reflection over the y-axis: g(x) = f(-x)
Reflection over y = x: g(x) and f(x) are swapped
Reflection over other lines (e.g., y = 1, x = -3) involves vertical or horizontal shifts

Theorems

Reflection Theorem: Reflecting a function over a line transforms the equation based on the mirror's symmetry

Suitable Grade Level

Grades 10-12

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