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Coordinate Geometry
and Graph Interpretation

Master the coordinate plane, distance and midpoint formulas, slopes, line equations, and graph interpretation: essential tools for GRE Quantitative success.

GRE Quantitative

The Coordinate Plane

Every point in the plane is described by an ordered pair \((x, y)\). The horizontal axis is the \(x\)-axis and the vertical axis is the \(y\)-axis. Their intersection is the origin \((0, 0)\).

\(x\)
\(y\)
\((0, 0)\)
I \((+, +)\)
II \((-, +)\)
III \((-, -)\)
IV \((+, -)\)
\(2\)
\(-2\)
\(2\)
\(-2\)

The Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) comes directly from the Pythagorean Theorem:

\((1, 1)\)
\((7, 5)\)
\(6\) units
\(4\) units
\(d\)

The Midpoint Formula

The midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is the average of the coordinates:

\((2, 1)\)
\((8, 5)\)
\((5, 3)\)

Slope: Rise over Run

The slope \(m\) measures how steep a line is. Given two points on the line:

Positive Slope
Line goes up from left to right. \(m > 0\)
Negative Slope
Line goes down from left to right. \(m < 0\)
Zero Slope
Horizontal line. \(m = 0\)

Equations of a Line

Slope-Intercept
\(y = mx + b\)
where \(m\) is the slope and \(b\) is the \(y\)-intercept.
Point-Slope
\(y - y_1 = m(x - x_1)\)
Uses a known point and the slope.
Standard Form
\(Ax + By = C\)
where \(A\), \(B\), \(C\) are integers.

Parallel vs. Perpendicular Lines

Parallel
same slope
Perpendicular
slopes are negative reciprocals
Parallel
\(m_1 = m_2\). The lines never meet.
Perpendicular
\(m_1 \cdot m_2 = -1\). They meet at a right angle.

Slope vs. \(y\)-Intercept

Slope \(m\) \(y\)-Intercept \(b\)
Measures Steepness / direction Where line crosses \(y\)-axis
Formula \(\frac{y_2 - y_1}{x_2 - x_1}\) Set \(x = 0\), solve for \(y\)
Sign tells you Up-right (+) or down-right (−) Above (+) or below (−) origin
Parallel lines Same \(m\) Different \(b\)

Step-by-Step Problem Strategy

1
Plot or identify key points. Read coordinates carefully from the graph or problem statement.
2
Choose the right formula. Distance, midpoint, slope, or line equation: match the formula to the question.
3
Substitute carefully. Watch signs: subtracting a negative coordinate means adding.
4
Simplify and check. Reduce fractions, simplify radicals, and verify with a quick sketch.

Example 1: Distance Between Two Points

Problem: Find the distance between \(A(-3, 2)\) and \(B(5, -4)\).

\(A(-3, 2)\)
\(B(5, -4)\)
\(8\)
\(6\)
\(d\)

Example 2: Finding the Equation of a Line

Problem: A line passes through \((1, 3)\) and \((4, 9)\). Write its equation in slope-intercept form.

1
Find the slope: \(m = \frac{9-3}{4-1} = \frac{6}{3} = 2\)
2
Use point-slope form: \(y - 3 = 2(x - 1)\)
3
Simplify: \(y = 2x - 2 + 3 = 2x + 1\)

Example 3: Reading a Graph

Problem: The graph below shows a line. Determine its slope and \(y\)-intercept.

\(1\)
\(2\)
\(-1\)
\(-2\)
\(1\)
\(2\)
\(-1\)
\((0, 2)\)
\((1, 1)\)
run = \(1\)
rise = \(-1\)

Finding \(x\) and \(y\) Intercepts

\(y\)-intercept
Set \(x = 0\) and solve for \(y\). The point is \((0, b)\).
\(x\)-intercept
Set \(y = 0\) and solve for \(x\). The point is \((a, 0)\).

Example: For the line \(3x + 4y = 12\):

y
\(y\)-intercept: \(3(0) + 4y = 12 \implies y = 3\). Point: \((0, 3)\)
x
\(x\)-intercept: \(3x + 4(0) = 12 \implies x = 4\). Point: \((4, 0)\)

Graphs of Quadratics

The graph of \(y = ax^2 + bx + c\) is a parabola. Its vertex is at \(x = -\frac{b}{2a}\). If \(a > 0\), it opens up; if \(a < 0\), it opens down.

\(x\)
\(y\)
\(1\)
\(2\)
\(-1\)
\(-2\)
\( \text{vertex } (0, 0) \)
\(y = x^2\)

Common Mistakes

✕ Sign Errors in Slope
Forgetting that subtracting a negative becomes addition. \(\frac{3 - (-2)}{4 - 1} = \frac{5}{3}\), not \(\frac{1}{3}\).
✕ Swapping \(x\) and \(y\)
The slope formula is \(\frac{\Delta y}{\Delta x}\), not \(\frac{\Delta x}{\Delta y}\). Rise is always vertical change.
✕ Forgetting to Square
In the distance formula, both differences must be squared before adding. Don't add first, then square.

Formula Reference

Concept Formula When to Use
Distance \(\sqrt{(\Delta x)^2 + (\Delta y)^2}\) Length of a segment
Midpoint \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) Center of a segment
Slope \(\frac{y_2 - y_1}{x_2 - x_1}\) Steepness, direction
Line Equation \(y = mx + b\) Equation from slope + point
Perpendicular \(m_1 \cdot m_2 = -1\) Right-angle relationship

Question 1 of 5

What is the distance between the points \((-1, 4)\) and \((3, 1)\)?

Question 2 of 5

What is the slope of a line that passes through \((2, -3)\) and \((6, 5)\)?

Question 3 of 5

A line has the equation \(y = 3x - 7\). A second line is perpendicular to it. What is the slope of the second line?

Question 4 of 5

What is the midpoint of the segment with endpoints \((-4, 6)\) and \((8, -2)\)?

Question 5 of 5

The vertex of the parabola \(y = 2x^2 - 8x + 3\) has an \(x\)-coordinate of: