linear functions and transformations of functions

Parent function

The simplest function in a function family (e.g., $f(x) = x^2$ or $f(x) = |x|$).

Transformation of a function

A change to a graph's position, direction, or size.

Four main function transformations

Translations, reflections, stretches, and compressions.

Parent function for a parabola

$$f(x) = x^2$$

Parent function for a V-shaped graph

$$f(x) = |x|$$

Transformation $f(x) + k$

Shifts the graph vertically up $k$ units if $k > 0$, or down $|k|$ units if $k < 0$.

Transformation $f(x) - k$

Shifts the graph down $k$ units (for $k > 0$).

Transformation $f(x - h)$

Shifts the graph right $h$ units (for $h > 0$).

Transformation $f(x + h)$

Shifts the graph left $h$ units (for $h > 0$).

Reason $f(x + h)$ shifts a graph left

Horizontal changes occur inside the function, making their effect opposite to visual intuition.

Rule for transformations outside $f(x)$

Outside changes are vertical: addition/subtraction shifts up or down, and multiplication alters vertical size.

Rule for transformations inside $f(x)$

Inside changes are horizontal and reversed: $x - h$ shifts right, while $x + h$ shifts left.

Transformation $-f(x)$

Reflects the graph across the $x$-axis.

Transformation $f(-x)$

Reflects the graph across the $y$-axis.

Effect of reflection across the $x$-axis on coordinates

Changes every $y$-value to its opposite, turning points $(x, y)$ into $(x, -y)$.

Effect of reflection across the $y$-axis on coordinates

Changes every $x$-value to its opposite, turning points $(x, y)$ into $(-x, y)$.

Reflection of $f(x) = x^2$ across the $y$-axis

Nothing visibly changes because $f(-x) = (-x)^2 = x^2$ due to symmetry about the $y$-axis.

Reflection of $f(x) = |x|$ across the $y$-axis

Nothing visibly changes because $|-x| = |x|$ due to symmetry about the $y$-axis.

Reflection of $f(x) = x^2$ across the $x$-axis

It becomes $-x^2$, creating a parabola that opens downward.

Reflection of $f(x) = |x|$ across the $x$-axis

It becomes $-|x|$, creating an upside-down V-shape.

Effect of $a \cdot f(x)$ when $a > 1$

Causes a vertical stretch by a factor of $a$, making the graph narrower and taller.

Effect of $a \cdot f(x)$ when $0 < a < 1$

Causes a vertical compression by a factor of $a$, making the graph wider and shorter.

Effect of a negative $a$ in $a \cdot f(x)$

Reflects the graph across the $x$-axis and scales its vertical size by $|a|$.

Comparison of $g(x) = 3x^2$ to $f(x) = x^2$

A vertical stretch by a factor of $3$.

Comparison of $g(x) = 0.25|x|$ to $f(x) = |x|$

A vertical compression by a factor of $0.25$ (or $\frac{1}{4}$).

Comparison of $g(x) = -2|x|$ to $f(x) = |x|$

A reflection across the $x$-axis and a vertical stretch by a factor of $2$.

Transformation $f(bx)$ when $b > 1$

Causes a horizontal compression by a factor of $\frac{1}{b}$.

Transformation $f(bx)$ when $0 < b < 1$

Causes a horizontal stretch by a factor of $\frac{1}{b}$.

Transformation $f(-bx)$

Includes a horizontal reflection across the $y$-axis and horizontal scaling by a factor of $\frac{1}{b}$.

General transformation form equation

$$g(x) = a \cdot f(b(x - h)) + k$$

Role of $h$ in $g(x) = a \cdot f(b(x - h)) + k$

Controls horizontal shift (right $h$ units when $h > 0$).

Role of $k$ in $g(x) = a \cdot f(b(x - h)) + k$

Controls vertical shift (up $k$ units when $k > 0$).

Role of $a$ in $g(x) = a \cdot f(b(x - h)) + k$

Controls vertical stretch/compression and reflection across the $x$-axis if negative.

Role of $b$ in $g(x) = a \cdot f(b(x - h)) + k$

Controls horizontal stretch/compression and reflection across the $y$-axis if negative.

Vertex of $y = (x - h)^2 + k$

$$(h, k)$$

Vertex of $y = a(x - h)^2 + k$

$$(h, k)$$

Vertex of $y = a|x - h| + k$

$$(h, k)$$

Vertex of $y = (x - 3)^2 - 6$

$$(3, -6)$$

Equation of parabola $f(x) = x^2$ shifted to vertex $(3, -6)$

$$g(x) = (x - 3)^2 - 6$$

Equation of parent parabola $f(x) = x^2$ shifted right $4$ and up $2$

$$g(x) = (x - 4)^2 + 2$$

Equation of parent parabola $f(x) = x^2$ shifted left $4$ and down $2$

$$g(x) = (x + 4)^2 - 2$$

Equation of $f(x) = |x|$ shifted right $3$ and down $5$

$$g(x) = |x - 3| - 5$$

Equation of $f(x) = |x|$ shifted left $2$ and up $1$

$$g(x) = |x + 2| + 1$$

Transformations converting $f(x) = x^2$ to $g(x) = (x - 3)^2 + 5$

Shift right $3$ units and up $5$ units.

Transformations converting $f(x) = x^2$ to $g(x) = (x + 1)^2$

Shift left $1$ unit.

Transformations converting $f(x) = |x|$ to $g(x) = |x| - 5$

Shift down $5$ units.

Transformations converting $f(x) = |x|$ to $g(x) = |x - 3|$

Shift right $3$ units.

Transformations converting $f(x) = x^2$ to $g(x) = x^2 + 2$

Shift up $2$ units.

Transformations converting $f(x) = x^2$ to $g(x) = -x^2 - 6$

Reflect across the $x$-axis, then shift down $6$ units.

Transformations converting $f(x) = x^2$ to $g(x) = 2(x + 1)^2$

Shift left $1$ unit and vertically stretch by a factor of $2$.

Transformations converting $f(x) = x^2$ to $g(x) = 4(x - 7)^2 - 9$

Shift right $7$ units, vertically stretch by a factor of $4$, then shift down $9$ units.

Difference between translation and reflection

A translation slides a graph without flipping it, whereas a reflection flips the graph over an axis.

Difference between vertical and horizontal stretch

A vertical stretch multiplies $y$-values, while a horizontal stretch modifies $x$-values using a factor inside the function.

Domain of $f(x) = x^2$

All real numbers: $(-\infty, \infty)$

Range of $f(x) = x^2$

$y \ge 0$, or $[0, \infty)$

Domain of $f(x) = |x|$

All real numbers: $(-\infty, \infty)$

Range of $f(x) = |x|$

$y \ge 0$, or $[0, \infty)$

Domain of $g(x) = (x - h)^2 + k$

All real numbers: $(-\infty, \infty)$

Range of $g(x) = a(x - h)^2 + k$ when $a > 0$

$$y \ge k$$

Range of $g(x) = a(x - h)^2 + k$ when $a < 0$

$$y \le k$$

Domain of $g(x) = a|x - h| + k$

All real numbers: $(-\infty, \infty)$

Range of $g(x) = a|x - h| + k$ when $a > 0$

$$y \ge k$$

Range of $g(x) = a|x - h| + k$ when $a < 0$

$$y \le k$$

Range of $g(x) = (x - 3)^2 - 6$

$$y \ge -6$$

Range of $g(x) = -(x - 3)^2 - 6$

$$y \le -6$$

Range of $g(x) = |x + 2| + 1$

$$y \ge 1$$

Vertex of $f(x) = |x + 2| + 1$

$$(-2, 1)$$

Formula for $f(-x)$ if $f(x) = |x + 2| + 1$

$|-x + 2| + 1$, which simplifies to $|x - 2| + 1$

Graph of $f(-x)$ given $f(x) = |x + 2| + 1$

Reflection of $f(x)$ across the $y$-axis, with vertex $(2, 1)$.

Error in stating $|2x - 4|$ shifts $|x|$ right $4$ units

Factor first: $|2x - 4| = |2(x - 2)|$. The shift is right $2$ units, not $4$ units.

Comparison of $g(x) = |2x - 4|$ to $f(x) = |x|$

Horizontal compression by a factor of $\frac{1}{2}$ and a shift right $2$ units.

Explanation for $2|x| = |2x|$

For every real $x$, multiplying the absolute value by $2$ gives the same output as evaluating $2x$ inside the absolute value: $2|x| = |2x|$.

$2|x|$ described as a vertical transformation

Vertical stretch of $|x|$ by a factor of $2$.

$|2x|$ described as a horizontal transformation

Horizontal compression of $|x|$ by a factor of $\frac{1}{2}$.

Conclusion drawn from $2|x| = |2x|$

Different transformation descriptions can produce identical graphs.

Procedure to find the $x$-intercept of a function

Set $y$ or $g(x) = 0$ and solve for $x$.

Procedure to find the $y$-intercept of a function

Set $x = 0$ and evaluate the function for $y$.

$x$-intercepts of $y = |x| - 5$

Set $0 = |x| - 5 \implies |x| = 5$. The intercepts are $(-5, 0)$ and $(5, 0)$.

$y$-intercept of $y = |x| - 5$

Set $x = 0 \implies y = |0| - 5 = -5$. The intercept is $(0, -5)$.

$x$-intercepts of $y = (x - 3)^2 - 9$

Set $0 = (x - 3)^2 - 9 \implies (x - 3)^2 = 9 \implies x = 0$ or $x = 6$. The intercepts are $(0, 0)$ and $(6, 0)$.

$y$-intercept of $y = (x - 3)^2 - 9$

Set $x = 0 \implies y = (-3)^2 - 9 = 0$. The intercept is $(0, 0)$.

First step when analyzing multiple transformations in an equation

Identify the parent function, then identify each change inside and outside the function.

Recommended order for applying multiple transformations

Horizontal scaling/reflection, horizontal shift, vertical scaling/reflection, then vertical shift.

Graphing procedure for a parabola in vertex form

Plot the vertex $(h, k)$, use the parent parabola shape, then apply any reflection or stretch/compression.

Graphing procedure for a translated absolute-value function

Plot the vertex $(h, k)$, use the V-shape from parent $|x|$, and apply any reflection or stretch/compression.

Meaning of $a > 0$ in $y = a(x - h)^2 + k$

The parabola opens upward.

Meaning of $a < 0$ in $y = a(x - h)^2 + k$

The parabola opens downward.

Meaning of $a > 0$ in $y = a|x - h| + k$

The V-shaped graph opens upward.

Meaning of $a < 0$ in $y = a|x - h| + k$

The V-shaped graph opens downward.