MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
Preparation for Calculus

P.1 Graphs and Models

Focus on intercepts, symmetry, graph transformations.

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Ideas in this lesson

01InterceptsWorked steps · five questions02SymmetryWorked steps · five questions03Graph transformationsWorked steps · five questions
Idea 1 of 3

Intercepts

Set y=0y = 0 for xx-intercepts and x=0x = 0 for the yy-intercept. Keep every real solution.

Worked example 1

Find the yy-intercept of y=x2−2y = x^{2} - 2.
  1. x=0x = 0
  2. y=02−2y = 0^{2} - 2
  3. y=−2y = -2

Answer: −2-2

Worked example 2

Find the positive xx-intercept of y=x2−9y = x^{2} - 9.
  1. 0=x2−90 = x^{2} - 9
  2. (x−3)(x+3)=0(x - 3)(x + 3) = 0
  3. x=±3x = \pm 3; positive root =3= 3

Answer: 33

Idea 2 of 3

Symmetry

Test f(−x)f(-x). Equality with f(x)f(x) gives yy-axis symmetry; equality with −f(x)-f(x) gives origin symmetry.

Worked example 1

Classify f(x)=x4+2f(x) = x^{4} + 2 as even, odd, or neither.
Coordinate graph-2-1.5-1-0.50.511.525101520xy
  1. f(−x)=x4+2f(-x) = x^{4} + 2
  2. f(−x)=f(x)f(-x) = f(x)
  3. f(−x)=f(x)  ⟹  f(-x) = f(x) \implies Even

Answer: Even

Worked example 2

Classify f(x)=x3−3xf(x) = x^{3} - 3x as even, odd, or neither.
Coordinate graph-2-1.5-1-0.50.511.525101520xy
  1. f(−x)=−x3+3xf(-x) = -x^{3} + 3x
  2. f(−x)=−f(x)f(-x) = -f(x)
  3. f(−x)=−f(x)  ⟹  f(-x) = -f(x) \implies Odd

Answer: Odd

Idea 3 of 3

Graph transformations

In a(x−h)2+ka(x - h)^{2} + k the vertex is (h,k)(h, k). The sign of a determines the opening direction.

Worked example 1

For f(x)=(x−2)2+1f(x) = (x - 2)^{2} + 1, give the vertex xx-coordinate.
  1. (x−2)2≥0(x - 2)^{2} \ge 0; (x−2)2=0(x - 2)^{2} = 0 ⇔ x=2x = 2
  2. Vertex =(2,1)= (2, 1)
  3. f(x)≥1f(x) \ge 1; requested value =2= 2

Answer: 22

Worked example 2

For f(x)=(x−3)2+2f(x) = (x - 3)^{2} + 2, give the minimum value.
  1. (x−3)2≥0(x - 3)^{2} \ge 0; (x−3)2=0(x - 3)^{2} = 0 ⇔ x=3x = 3
  2. Vertex =(3,2)= (3, 2)
  3. f(x)≥2f(x) \ge 2; requested value =2= 2

Answer: 22

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §P.1,pP.1, p. 22. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.