ORIGINAL WORKED EXAMPLE
Substitute an expression
If f(x)=x²−3x, find f(t+2).
- Replace each x by t+2: (t+2)²−3(t+2).
- Expand: t²+4t+4−3t−6.
- Combine to obtain t²+t−2. f(t+2) is not f(t)+2.
Answer t²+t−2
GRE · Algebra · TOPIC LESSON
Functions, graphs, slopes and intercepts
Connect formulas, coordinate positions and graph features while keeping domains and scales explicit.
BY THE END, YOU CAN
- Evaluate and compose functions, including piecewise definitions.
- Find slopes, intercepts and intersections.
- Interpret linear, quadratic and absolute-value graphs and shaded inequalities.
BEFORE YOU START
- Algebra and equations
- Number lines and distance
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
Original GRE-style worked examples and practice
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
Coordinates locate a point in two directions
An ordered pair (x,y) gives horizontal position first and vertical position second. The axes meet at the origin (0,0). Points on the x-axis have y=0; points on the y-axis have x=0. A labeled scale can use different units on the two axes, so visible steepness alone does not give numerical slope.
GRE coordinate axes and data graphs are drawn to scale, unlike ordinary geometry sketches. Read tick labels and units. The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²], and their midpoint averages the two coordinates separately.
02 / UNDERSTAND & APPLY
A function assigns one output to each allowed input
f(x) names the output of a function at input x; it is not f multiplied by x. Substitute the whole input expression into every x. The domain contains allowed inputs; the range contains resulting outputs. Denominators exclude zero and real square roots require nonnegative radicands.
In a piecewise function, the input chooses the branch. Apply only the formula whose condition contains that input, especially at a boundary. A function can assign the same output to different inputs; it cannot assign two outputs to one input.
03 / UNDERSTAND & APPLY
The inside function acts first
For f(g(x)), first calculate g(x), then use its output as the input to f. The order usually matters. The input must belong to g’s domain, and g(x) must belong to f’s domain.
An inverse function, when it exists, reverses the input-output assignment; it is not the reciprocal 1/f(x). For a one-to-one linear function f(x)=ax+b with a≠0, solving y=ax+b for x gives inverse (y−b)/a.
ORIGINAL WORKED EXAMPLE
Composition in both orders
Let f(x)=2x−1 and g(x)=x². Find f(g(3)) and g(f(3)).
- g(3)=9, so f(g(3))=f(9)=17.
- f(3)=5, so g(f(3))=g(5)=25.
- The different answers show why composition order cannot be swapped.
Answer 17 and 25
04 / UNDERSTAND & APPLY
Slope is change in output per change in input
For two points with distinct x-coordinates, slope m=(y₂−y₁)/(x₂−x₁). Keep subtraction order consistent in numerator and denominator. A positive slope rises left to right; a negative slope falls. Horizontal lines have slope zero; vertical lines have undefined slope.
Parallel nonvertical lines have equal slopes. Perpendicular nonvertical lines with nonzero slopes have product −1; a horizontal and vertical pair is perpendicular too. Numerical slope depends on axis units, not just the angle a sketch appears to show.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
Find a line from two points
Find the equation through (−2,−3) and (4,9).
- Slope=(9−(−3))/(4−(−2))=12/6=2.
- Use y=2x+b with (4,9): 9=8+b, so b=1.
- The line is y=2x+1; substituting (−2,−3) checks the other point.
Answer y=2x+1
05 / UNDERSTAND & APPLY
Set the appropriate coordinate to zero
For y=mx+b, the y-intercept is (0,b). Find an x-intercept by setting y=0 and solving; if m≠0, its x-coordinate is −b/m. A graph may have several x-intercepts, no x-intercepts, or infinitely many points on an axis.
To find where two graphs intersect, set their y-expressions equal and solve for x, then calculate y. Two line graphs may intersect once, be parallel, or coincide. A line and parabola may intersect at zero, one or two real points.
ORIGINAL WORKED EXAMPLE
Intersection and intercept are different tasks
Find the intersection of y=2x+1 and y=−x+7.
- Equate outputs: 2x+1=−x+7.
- 3x=6, so x=2; substitute to get y=5.
- The intersection is (2,5), not either line’s y-intercept.
Answer (2,5)
06 / UNDERSTAND & APPLY
Recognize shape and transformations
The graph y=|x| is a V with vertex at (0,0). y=|x−h|+k shifts the vertex to (h,k); a negative coefficient reflects it downward. The graph y=(x−h)²+k is an upward parabola with vertex (h,k); multiplying the square by a negative number makes it open downward.
Do not confuse a horizontal shift’s sign: x−3 places the vertex at x=3. Intercepts and sampled points provide checks. A sketch must preserve the declared coordinate scale if used to compare positions.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
Read an absolute-value graph algebraically
For y=|x−2|−1, find the vertex and x-intercepts.
- The vertex occurs when x−2=0, giving (2,−1).
- At an x-intercept, 0=|x−2|−1, so |x−2|=1.
- x=1 or x=3, giving (1,0) and (3,0).
Answer Vertex (2,−1); x-intercepts (1,0) and (3,0)
07 / UNDERSTAND & APPLY
A boundary plus a tested side gives the solution region
For y>mx+b, use the boundary line y=mx+b and the region above it; the boundary itself is excluded. For y≥mx+b, include the boundary. A vertical inequality such as x<3 describes points to the left of a vertical boundary.
Test a point not on the boundary when uncertain. For a system, intersect regions; do not keep all points from either region indiscriminately. Reading a graph gives approximate values unless exact coordinates or equations justify precision.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
08 / UNDERSTAND & APPLY
Check what the graph or model actually represents
A physical model can restrict a function’s mathematical domain: a cost formula may permit only nonnegative time or whole-item inputs. An apparent line segment on a graph does not authorize extrapolation beyond its stated interval. A graph of paired data does not automatically define a function over every intermediate input.
Calculate distances and slopes from coordinates rather than measuring pixels. A distance formula combines horizontal and vertical lengths in the same physical unit; mismatched units need conversion before geometric distance has meaning.
09 / UNDERSTAND & APPLY
Choose a method and answer the requested question
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
10 / UNDERSTAND & APPLY
Original practice: read inputs, axes and boundaries
Use exact coordinates where the equations supply them.
ORIGINAL PRACTICE · 1
Numeric entry: f(x)=3x²−2. Find f(−2).
Reveal answer & reasoning
10
f(−2)=3·4−2=10. Squaring a parenthesized negative input produces a positive value.
ORIGINAL PRACTICE · 2
Single answer: The slope through (1,5) and (4,−1) is A −2; B −1/2; C 1/2; D 2; E 6.
Reveal answer & reasoning
A
Slope=(−1−5)/(4−1)=−6/3=−2. Swapping rise and run gives the reciprocal, not slope.
ORIGINAL PRACTICE · 3
Numeric entry: What is the x-coordinate of the x-intercept of 3x+2y=12?
Reveal answer & reasoning
4
Set y=0, giving 3x=12. Setting x=0 instead would locate the y-intercept.
ORIGINAL PRACTICE · 4
Select all that apply: Which points lie on or above y=2x−1? A (0,0); B (1,0); C (2,3); D (−1,−4); E (3,6).
Reveal answer & reasoning
A, C and E
Substitute each pair into y≥2x−1. A gives 0≥−1, C 3≥3, and E 6≥5. B gives 0<1; D gives −4<−3. Boundary equality is allowed.
ORIGINAL PRACTICE · 5
Numeric entry: What is the distance between (−1,2) and (5,10), with both axes in centimeters?
Reveal answer & reasoning
10
Changes are 6 and 8 cm. Distance=√(36+64)=10 cm. Adding the changes gives a path along two perpendicular segments, not the straight-line distance.
ORIGINAL PRACTICE · 6
QC: f(x)=x². Quantity A: f(−3). Quantity B: f(3).
Reveal answer & reasoning
C
Both outputs are 9. A function may map different inputs to the same output.
ORIGINAL PRACTICE · 7
Numeric entry: f(x)=x+5 when x<2, and f(x)=3x when x≥2. Find f(2).
Reveal answer & reasoning
6
The input 2 belongs to the second branch because its inequality includes equality. Apply 3·2=6. The excluded first branch would give 7, but its condition x<2 is false at 2; do not apply both formulas or average them.
ORIGINAL PRACTICE · 8
Single answer: Which is the vertex of y=|x+4|+2? A (4,2); B (−4,2); C (−4,−2); D (2,4); E (0,6).
Reveal answer & reasoning
B
The absolute-value term is zero when x=−4, giving y=2. The point (0,6) is on the graph but is not its vertex.
KEEP THE BIG IDEAS
Quick revision
- A function input must be substituted everywhere and remain in the domain.
- Slope is rise divided by run with consistent subtraction order.
- Intersections satisfy both equations; intercepts set one coordinate to zero.
- Use graph scales and inclusion rules rather than visual guesswork.
READ WITH CONTEXT
References & scope
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