y-axis as mirror/reflection in y-asis

$\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ → $\left(\begin{array}{c}\mathrm{-x}\\ \mathrm{y}\end{array}\right)$

Thus the x is negated.

$\left(\begin{array}{c}-2\\ 2\end{array}\right)$ → $\left(\begin{array}{c}\mathrm{-(-2)}\\ 2\end{array}\right)$ → $\left(\begin{array}{c}2\\ 2\end{array}\right)$ → (2,2)

Gradient = $\frac{\mathrm{Change\; in\; Y}}{\mathrm{Change\; in\; X}}$ = $\frac{{\mathrm{Y}}_2-{\mathrm{Y}}_1}{{\mathrm{X}}_2-{\mathrm{X}}_1}$

Y₂ = -7
Y₁ = -3
X₂ = -4
X₁ = -5

Gradient = $\frac{\mathrm{-7-(-3)}}{\mathrm{-4-(-5)}}$ = $\frac{\mathrm{-7+3}}{\mathrm{-4\; +\; 5}}$ = $\frac{-4}{1}$ = -4

Let $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ = the translation vector.

$\left(\begin{array}{c}2\\ 6\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ = $\left(\begin{array}{c}5\\ 4\end{array}\right)$

$\left(\begin{array}{c}\mathrm{2\; +\; x}\\ \mathrm{6\; +\; y}\end{array}\right)$ = $\left(\begin{array}{c}5\\ 4\end{array}\right)$

The top at the left equals the top at the right and the bottom at the left equals the bottom at the right.

2 + x = 5
x = 5 - 2
x = 3

6 + y = 4
y = 4 - 6
y = -2

$\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ = $\left(\begin{array}{c}3\\ -2\end{array}\right)$

Reflection in axis

Reflection in the x axis

The rule for a reflection over the x -axis is (x, y)→(x, −y).

In the reflection in the x, y is negative

Reflection in the y axis

The rule for a reflection over the y -axis is (x, y)→(−x, y).

Reflection in the line y=x :

The rule for a reflection in the line y=x is (x, x)→(y, x)

Reflection in the line y=−x :

The rule for a reflection in the origin is (x, y)→(−y, −x) .

Reflection of the point (x, y) = (-x, y)

Reflection of (4, 3) in the y-axis → S^' = (-4, 3)

Gradient of a line = change in y / change in x =(y₂-y₁)/(x₂-x₁)

The general point of a line is in the form (x,y), thus the first cordinate is x and the second cordinate is y

P(3,-2) and Q(-2,4)

y₂ = 4 and y₁ = -2

x₂ = -2 and x₁ = 3

Gradient = (4--2)/(-2-3)

-- = +

Gradient = (4+2)/(-2-3) = ⁶⁄_-5 = -⁶⁄₅

Property: Rotations by Multiples of 90 Degrees about the Origin

For a point with coordinates (𝑥,𝑦),the following is true:

From point 4, rotation of point through 360^o about the origin is the same point, hence the image of rotating the point (-3, 5) through 360^o will be the same, (-3, 5)

Thus the difference of y coordinates over the x coordinates

A(-3,5) B(7,-2)

For a point (x,y) the first coordinate is x and the second is y

y2 = -2, y1 = 5, x2 = 7, x1 = -3

The rule for y -axis reflection is (x,y)→(−x,y), thus reflection in the y-axis, the x is negated.

⇒ (-2,3) reflected in the y-axis will be -(-2),3 but -- = +, hence (2,3)

A reflection is a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane. When reflecting a figure in a line or in a point, the image is congruent to the preimage.

A reflection maps every point of a figure to an image across a fixed line. The fixed line is called the line of reflection.

Some simple reflections can be performed easily in the coordinate plane using the general rules below.

Reflection in the x -axis:

A reflection of a point over the x -axis is shown.

The rule for a reflection over the x -axis is (x,y)→(x,−y), thus reflection in the x-axis, the y is negated.

Reflection in the y -axis:

A reflection of a point over the y -axis is shown.

The rule for a reflection over the y -axis is (x,y)→(−x,y), thus reflection in the y-axis, the x is negated.

Reflection in the line y=x :

A reflection of a point over the line y=x is shown.

The rule for a reflection in the line y=x is (x,y)→(y,x), thus you simply switch their positions.

Reflection in the line y=−x :

A reflection of a point over the line y=−x is shown.

The rule for a reflection in the origin is (x,y)→(−y,−x), thus you simply negate the result of y = x.

Equation of a straight line, y = mx+c

where m is the gradient of the line

m = change in y / change in x =(y₂-y₁)/(x₂-x₁)

NOTE: The first coordinate represents the x-axis and the second the y-axis

Thus in (-3,5), x = -3 and y = 5

In (6,8), x = 6 and y = 8

⇒ The gradient of the line = (8-5)/(6--3) = 3/(6+3) = 3/9 = 1/3

Use any of the points and the gradient to calculate the value of c

Using the point (-3,5)

⇒ 5 = ⅓ x -3 + c

⇒ 5 = -1 + c

⇒ 5 + 1= c

⇒ c = 6

Using the point (6,8)

⇒ 8 = ⅓ x 6 + c

⇒ 8 = 2 + c

⇒ 8 - 2= c

⇒ c = 6

Hence the equation will be y = ⅓^x+6

Translation

A translation moves a shape up, down or from side to side but it does not change its appearance in any other way.

Translation is an example of a transformation. A transformation is a way of changing the size or position of a shape.

Every point in the shape is translated the same distance in the same direction.

If a point (x,y) is translated by the vector ( ^a _b), the image formed will have the point (x+a , y+b)

From the question the translation vector is (^-2 ₃), the image point is P¹ = (4,-5)

⇒ (4,-5) = (x+-2,y+3)

⇒ for the x coordinate, 4 = x-2

⇒ for the y coordinate, -5 = y+3

Solve for the values of x and y

4 = x-2

4+2 = x ⇒ x = 6

-5 = y+3

-5 - 3 = y ⇒ y = -8

⇒ The point translated = (6,-8)