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Mastering Graphs
on the SAT

Learn to read, interpret, and analyze graphs of linear, quadratic, and other functions: the skill that connects algebra to visual reasoning.

SAT Math

What Is a Graph?

A graph is a visual representation of all the ordered pairs \((x, y)\) that satisfy an equation. Every point on the curve is a solution, and every solution is a point on the curve.

x y (1, 1) (2, 2) (3, 3) \(y=x\)

The Coordinate Plane

The SAT places every graph on the \(xy\)-coordinate plane. You need to quickly identify axes, scale, intercepts, and the direction of a curve.

\(x\)-axis
The horizontal axis: independent variable (input).
\(y\)-axis
The vertical axis: dependent variable (output).
Origin
The point \((0, 0)\) where both axes meet.

Why Graphs Matter on the SAT

Graph questions appear in multiple forms across both SAT Math modules. They test whether you can translate between equations and pictures.

1
Read values: Find \(f(3)\) from a graph, or determine where \(f(x) = 0\).
2
Match equations: Identify which equation corresponds to a given graph.
3
Interpret features: Slope, intercepts, vertex, axis of symmetry, and more.
4
Transformations: Shifts, reflections, and stretches applied to a parent function.

Key Features to Identify

Intercepts
Where the graph crosses the \(x\)-axis (\(y=0\)) and \(y\)-axis (\(x=0\)).
Slope
Rate of change: \(\frac{\text{rise}}{\text{run}}\). Positive slopes go up-right; negative go down-right.
Vertex
The turning point of a parabola: highest or lowest \(y\)-value.
Symmetry
Parabolas are symmetric about \(x = h\). Even functions are symmetric about the \(y\)-axis.

Slope-Intercept Form: \(y = mx + b\)

x y 1 2 3 4 1 2 b = 1 rise run \(m = \frac{2}{3}\)

Quadratic Graphs: Parabolas

x y 2 -2 4 -4 1 2 3 4 5 vertex (0, 0) axis of symmetry \(y = \tfrac{1}{2}x^2\)

Linear vs Quadratic Graphs

Linear Quadratic
Shape Straight line Parabola (U or inverted U)
Equation \(y = mx + b\) \(y = ax^2 + bx + c\)
Slope Constant Changes at every point
\(x\)-intercepts At most 1 0, 1, or 2
Key feature Slope & \(y\)-intercept Vertex & axis of symmetry

Reading Intercepts from a Graph

x y 1 2 3 4 5 1 2 3 y-int: (0, 3) x-int: (3, 0)

Step-by-Step: Analyzing a Graph

1
Read the axes: check variable names, units, and scale markings.
2
Identify the shape: line, parabola, or something else?
3
Find intercepts: where does the graph cross each axis?
4
Determine slope or vertex: for lines, compute \(\frac{\text{rise}}{\text{run}}\). For parabolas, locate the vertex.
5
Match to the equation: use the features you found to eliminate wrong answer choices.

Example 1: Finding Slope

A line passes through the points \((1, 2)\) and \((4, 8)\). What is its slope?

x y 1 2 3 4 5 1 2 3 4 5 6 7 (1, 2) (4, 8) rise = 6 run = 3

Example 2: Matching a Parabola

A parabola opens downward, has vertex \((2, 5)\), and crosses the \(x\)-axis at \(x = 0\) and \(x = 4\). Which equation matches?

x y 1 2 3 4 1 2 3 4 5 (2, 5) (0, 0) (4, 0)

Example 3: Systems on a Graph

Two lines are graphed. Where they intersect is the solution to the system. If the lines are parallel, there is no solution.

x y 1 2 3 4 5 1 2 3 \(y=x\) \(y=-x+4\) (2, 2)

Graph Transformations

The SAT tests four basic transformations of \(y = f(x)\):

\(f(x) + k\)
Shifts the graph up by \(k\) units (\(k > 0\)).
\(f(x - h)\)
Shifts the graph right by \(h\) units (\(h > 0\)).
\(-f(x)\)
Reflects the graph across the \(x\)-axis.
\(f(-x)\)
Reflects the graph across the \(y\)-axis.

Common Mistakes with Graphs

✕ Ignoring the Scale
Reading \(x = 3\) as \(x = 6\) because each gridline represents 2 units, not 1.
✕ Reversing Shift Direction
\(f(x - 3)\) shifts right, not left. Students often move the graph in the wrong direction.
✕ Slope Sign Error
Computing \(\frac{y_2 - y_1}{x_1 - x_2}\) instead of \(\frac{y_2 - y_1}{x_2 - x_1}\), flipping the sign of the slope.
✕ Confusing Roots and \(y\)-Intercept
Stating the \(y\)-intercept is where the graph crosses the \(x\)-axis. It is where \(x = 0\).

Graph Features at a Glance

Feature How to Find SAT Use
Slope \(\frac{y_2 - y_1}{x_2 - x_1}\) Rate of change, direction
\(y\)-intercept Set \(x = 0\), read \(y\) Starting value, \(b\) in \(y = mx + b\)
\(x\)-intercept(s) Set \(y = 0\), read \(x\) Zeros, roots, solutions
Vertex Turning point of parabola Max or min value
Intersection Where two graphs cross Solution to a system

Question 1 of 5

A line has a \(y\)-intercept of 4 and passes through the point \((2, 10)\). What is the slope of this line?

Question 2 of 5

The graph of \(y = a(x - 1)^2 + 3\) opens downward and has a vertex at \((1, 3)\). Which must be true about \(a\)?

Question 3 of 5

The graph of \(y = f(x)\) is shifted 3 units to the right and 2 units up. Which expression represents the new graph?

Question 4 of 5

Two lines intersect at the point \((3, -1)\). This point represents:

Question 5 of 5

A quadratic function has \(x\)-intercepts at \(x = -2\) and \(x = 6\). What is the \(x\)-coordinate of its vertex?