Math 2.5-2.8 Quiz

Transformation represented by $y = f(x) + a$

A vertical translation by $a$ units (upwards if $a > 0$, downwards if $a < 0$).

Transformation represented by $y = f(x - a)$

A horizontal translation by $a$ units (to the right if $a > 0$, to the left if $a < 0$).

Vector representation and description for transforming $y = x^2$ into $y = (x - 3)^2 + 2$

Translation vector $(3, 2)$, representing a translation $3$ units to the right and $2$ units up.

Transformation equation and coordinate change for a reflection of $y = f(x)$ in the $x$-axis

Equation: $y = -f(x)$ Coordinate change: $(x, y) \rightarrow (x, -y)$

Transformation equation and coordinate change for a reflection of $y = f(x)$ in the $y$-axis

Equation: $y = f(-x)$ Coordinate change: $(x, y) \rightarrow (-x, y)$

If $f(x) = x^2 + 2x - 3$, algebraic expressions for $-f(x)$ and $f(-x)$

$-f(x) = -x^2 - 2x + 3$$f(-x) = x^2 - 2x - 3$

Transformation represented by $y = af(x)$ and its effect on coordinates

Vertical stretch by scale factor $a$, transforming $(x, y)$ to $(x, ay)$.

Transformation represented by y = f\n\n\n\n\left(\frac{x}{a}\right) and its scale factor

Horizontal stretch by scale factor $a$, transforming $(x, y)$ to $(ax, y)$.

Scale factor and effect of the transformation $y = f(ax)$ when $a > 1$

Horizontal stretch by scale factor \n\n\frac{1}{a} (the graph is compressed horizontally).

Horizontal scale factors for $y = f(3x)$ and $y = f\left(\frac{x}{4}\right)$

For $y = f(3x)$, horizontal scale factor is $\frac{1}{3}$. For $y = f\left(\frac{x}{4}\right)$, horizontal scale factor is $4$.

General coordinate transformation of point $(u, v)$ on $y = f(x)$ under $y = af(b(x - h)) + k$

$$(u, v) \rightarrow \left(h + \frac{u}{b}, av + k\right)$$

Transformed position of point $(2, 5)$ under $y = -2f(x - 3) + 1$

$$(5, -9)$$

Horizontal shift: $2 + 3 = 5$ Vertical shift and stretch: $-2(5) + 1 = -9$

Function equation resulting from reflecting $y = f(x)$ in the $y$-axis, then stretching horizontally by scale factor $2$

$$y = f\left(-\frac{x}{2}\right)$$

Turning point of $y = x^2 + 6x + 8$ determined by completing the square

$$(-3, -1)$$

Completed square form: $y = (x + 3)^2 - 1$

Procedure for finding the $x$-intercepts and $y$-intercept of a transformed curve

Find $x$-intercepts by setting $y = 0$ and solving for $x$. Find the $y$-intercept by setting $x = 0$ and evaluating $y$.

Direction of movement for the graph $y = f(x - 3)$

Moves $3$ units to the right (not to the left).

Difference between $-f(x)$ and $f(-x)$ in graph reflection

$-f(x)$ reflects the graph in the $x$-axis. $f(-x)$ reflects the graph in the $y$-axis.