Math Problem Statement

The graph of β„Ž is obtained when 𝑓(π‘₯) =βˆ’ 3π‘₯ is reflected along the 2 + 12 line 𝑦 = 0. Write down the equation of β„Ž in the form β„Ž(π‘₯) = 𝑝π‘₯ 2 + π‘˜.

Solution

Let's break down the problem step-by-step:

  1. Given Function: We start with f(x)=βˆ’3xf(x) = -3x.

  2. Reflection Line: The reflection line is y=0y = 0, which is the x-axis.

  3. Reflection of f(x)f(x) Along the x-axis:

    • When we reflect f(x)=βˆ’3xf(x) = -3x across the x-axis, we change the sign of f(x)f(x).
    • The reflection of f(x)=βˆ’3xf(x) = -3x across y=0y = 0 is h(x)=3xh(x) = 3x.
  4. Writing h(x)h(x) in the Given Form:

    • We want h(x)h(x) in the form h(x)=px2+kh(x) = px^2 + k.
    • Since h(x)=3xh(x) = 3x is a linear function, p=0p = 0 and h(x)h(x) simplifies to a constant term, h(x)=3xh(x) = 3x.

Final Answer: [ h(x) = 3x

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Math Problem Analysis

Mathematical Concepts

Algebra
Reflections
Linear Functions

Formulas

Reflection of a function: y = -f(x) when reflected across the x-axis

Theorems

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Suitable Grade Level

Grades 9-10