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Zorluk: ZorCoordinate Geometry: Transformations and Geometric Graphs

In the xyxy-plane, line L1L_1 passes through the origin (0,0)(0,0) and the point (4,3)(4, 3). Line L2L_2 is formed by reflecting line L1L_1 across the vertical line x=2x = 2 and then translating the resulting line downward by 55 units. If line L2L_2 intersects the yy-axis at the point (0,k)(0, k), what is the value of kk?

  1. A
    −5-5
  2. −2-2Cevap
  3. C
    13\frac{1}{3}
  4. D
    33
  5. E
    88

Cevap

The correct value of kk is −2-2.
Line L1L_1 has slope 34\frac{3}{4} and equation y=34xy = \frac{3}{4}x. Reflecting across x=2x = 2 replaces xx with 4−x4 - x, transforming the equation into y=34(4−x)=3−34xy = \frac{3}{4}(4 - x) = 3 - \frac{3}{4}x. Translating downward by 55 units gives y=−34x−2y = -\frac{3}{4}x - 2. Setting x=0x = 0 gives the yy-intercept (0,−2)(0, -2), making −2-2 the correct value.

Adım Adım Çözüm

1
Find the equation of line L1L_1.
The slope of line L1L_1 passing through (0,0)(0,0) and (4,3)(4,3) is m=3−04−0=34m = \frac{3 - 0}{4 - 0} = \frac{3}{4}. Thus, the equation is y=34xy = \frac{3}{4}x.
Establishing the initial linear equation is necessary before applying coordinate transformations.
2
Apply the reflection across the line x=2x = 2.
Reflecting any point (x,y)(x, y) across x=2x = 2 transforms its x-coordinate to 2(2)−x=4−x2(2) - x = 4 - x. Substituting 4−x4 - x into the equation gives y=34(4−x)=3−34xy = \frac{3}{4}(4 - x) = 3 - \frac{3}{4}x.
Reflection across a vertical line x=ax = a preserves the y-values while mapping x↦2a−xx \mapsto 2a - x.
3
Apply the downward vertical translation by 55 units.
Subtracting 55 from the equation yields y=(3−34x)−5=−34x−2y = \left(3 - \frac{3}{4}x\right) - 5 = -\frac{3}{4}x - 2.
Translating a graph downward by cc units subtracts cc from the output yy.
4
Determine the y-intercept of line L2L_2.
Setting x=0x = 0 in y=−34x−2y = -\frac{3}{4}x - 2 yields y=−2y = -2, so k=−2k = -2.
The y-intercept of a line occurs where x=0x = 0.

Anahtar Kavram

Coordinate Transformations of Lines: Reflection across x=ax = a and Vertical Translations
Tahmini Süre:2m 0s
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Konular

Coordinate Geometry: Transformations and Geometric Graphs

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