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Graphing
Functions

Master the art of reading, interpreting, and sketching graphs of functions: key features, transformations, and real SAT strategies.

SAT Math

What Is a Function?

A function is a rule that assigns exactly one output \( y \) to each input \( x \). We write \( y = f(x) \). Its graph is the set of all points \( (x, f(x)) \) plotted on the coordinate plane.

x y 1 intersection = Function ✓

Key Features of a Graph

Intercepts
Where the graph crosses the axes: x-intercepts (set \( y=0 \)) and y-intercept (set \( x=0 \)).
Domain & Range
Domain: all valid \( x \)-values. Range: all resulting \( y \)-values you can read from the graph.
Increasing / Decreasing
A function increases when it rises left to right and decreases when it falls.
Max & Min
The highest and lowest points on the graph (vertices, turning points).

Why Graphing Matters on the SAT

The SAT tests your ability to read graphs, match equations to graphs, and identify transformations. You will not always need to compute: instead, you must reason visually.

1
Identify features: find intercepts, vertex, symmetry from a graph.
2
Match equation to graph: given 4 equations, pick the one whose graph is shown.
3
Apply transformations: shifts, reflections, stretches change a graph's shape and position.

The Parent Functions

Every function family has a parent function: the simplest form. On the SAT, the most common are:

Linear
\( f(x) = x \)
Straight line through origin
Quadratic
\( f(x) = x^2 \)
U-shaped parabola
Absolute Value
\( f(x) = |x| \)
V-shaped graph
x y 1 2 3 -1 -2 -3 1 2 3 4 \( y = x \) \( y = x^2 \) \( y = |x| \)

Transformation Rules

Starting from \( y = f(x) \), every transformation follows a predictable pattern:

Equation Transformation Direction
\( f(x) + k \) Shift up \( k \) Vertical
\( f(x) - k \) Shift down \( k \) Vertical
\( f(x - h) \) Shift right \( h \) Horizontal
\( f(x + h) \) Shift left \( h \) Horizontal
\( -f(x) \) Reflect over x-axis Vertical
\( f(-x) \) Reflect over y-axis Horizontal
\( a \cdot f(x) \), \( a > 1 \) Vertical stretch Vertical

Shifts vs. Reflections

Shifts (Translations)
Move the entire graph without changing its shape. Adding/subtracting inside \( f() \) shifts horizontally; outside shifts vertically.
Reflections
Flip the graph across an axis. A negative outside \( -f(x) \) flips over the x-axis. A negative inside \( f(-x) \) flips over the y-axis.

How to Analyze a Graph

1
Identify the parent function. Is it linear, quadratic, absolute value, or exponential?
2
Find the intercepts. Where does the graph cross the x-axis and y-axis?
3
Locate the vertex or key point. For parabolas, find the vertex \( (h, k) \).
4
Check transformations. Is it shifted, reflected, or stretched compared to the parent?
5
Verify with a point. Plug one visible point into each answer choice to eliminate options.

Example 1: Reading a Parabola

x y 1 2 3 -1 -2 -3 1 2 3 4 (1, -1) x-int: (0, 0) (2, 0)

Example 2: Identifying Transformations

Problem: The graph of \( g(x) = -(x+2)^2 + 4 \) is obtained from \( f(x) = x^2 \). Describe all transformations.

1
Identify the parent: \( f(x) = x^2 \) (upward parabola).
2
The \( (x+2) \) inside: shift left 2 units.
3
The negative sign: reflect over the x-axis (opens downward).
4
The \( +4 \) outside: shift up 4 units.

Example 3: Matching Equation to Graph

Problem: Which equation matches the graph below?

x y 1 2 -1 -2 1 2 3 (-1, -1) (-2, 0) (0, 0)

Example 4: Reading \( f(x) \) from a Graph

Problem: The graph of \( y = f(x) \) is shown. What is \( f(2) \)?

x y 1 2 3 4 -1 1 2 3 -1 \( f(2) = 3 \)

Vertex Form of a Quadratic

\( a > 0 \)
Parabola opens upward. Vertex is a minimum.
\( a < 0 \)
Parabola opens downward. Vertex is a maximum.
\( |a| > 1 \)
Parabola is narrower (vertical stretch).
\( |a| < 1 \)
Parabola is wider (vertical compression).

Common Mistakes

✕ Shift Direction Reversed
Thinking \( f(x - 3) \) shifts left instead of right. Remember: horizontal shifts are opposite to the sign inside.
✕ Confusing \( f(x) + k \) and \( f(x + k) \)
Adding outside \( f \) shifts vertically. Adding inside shifts horizontally. Don't mix them up.
✕ Forgetting to Check a Point
When matching an equation to a graph, always plug in a visible point to verify. Don't rely on shape alone.

Summary of Graph Features

Feature How to Find It SAT Usage
x-intercepts Set \( y = 0 \), solve Roots / solutions
y-intercept Set \( x = 0 \), evaluate Starting value
Vertex \( (h, k) \) from vertex form Max / min value
Axis of symmetry \( x = h \) Line of reflection
Direction Sign of \( a \) Opens up / down
Domain All valid \( x \)-values Input restrictions
Range All possible \( y \)-values Output restrictions

Question 1 of 5

The graph of \( y = f(x) \) has a vertex at \( (3, -2) \) and opens upward. Which could be the equation?

Question 2 of 5

The graph of \( g(x) = f(x) + 5 \) is obtained from the graph of \( f(x) \). What transformation occurred?

Question 3 of 5

If the graph of \( y = x^2 \) is reflected over the x-axis and then shifted right 4 units, which equation results?

Question 4 of 5

A function \( f \) is graphed on the coordinate plane. The graph passes through \( (0, 3) \) and \( (2, 7) \). What is \( f(2) - f(0) \)?

Question 5 of 5

The graph of \( y = |x| \) is shifted left 3 units and down 2 units. What is the vertex of the new graph?