Meet the function families

A straight line has constant slope m and vertical intercept b. Positive slope rises; negative slope falls. When m = 0, the function is constant. Its domain is all real numbers.

Change m to tilt the line and b to move its y-intercept. The marked point is (0, b).

f(-1)

-1

f(0)

1

f(1)

3

For f(x) = 2x + 1, find f(2).

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✓2 × 2 + 1 = 5. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

Try the linear prediction activity

A quadratic graph is a parabola. It opens upward when a > 0 and downward when a < 0. Its vertex is a minimum or maximum. The domain is all real numbers; x² has range [0, ∞).

Move the vertex along the x-axis, or flip the opening. Here f(x) = ±(x − h)²; the dashed line x = h is its axis of symmetry.

f(-1)

1

f(0)

0

f(1)

1

For f(x) = x², find f(−3).

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✓(−3)² = 9. Squaring a negative number gives a positive result. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

A polynomial adds powers of x with nonnegative integer exponents. The highest power with a nonzero coefficient is its degree. Linear and quadratic functions are also polynomials. All real inputs are allowed.

Explore x³ − cx. With c > 0, two turning points appear. The positive cubic term keeps the left end falling and the right end rising.

f(-1)

0

f(0)

0

f(1)

0

For f(x) = x³ − x, find f(2).

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✓2³ − 2 = 8 − 2 = 6. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

A rational function is a quotient of polynomials. Exclude every input that makes the original denominator zero, even if a factor cancels. For 1/x, zero is excluded and the graph has two separate branches.

Shift 1/(x − h) horizontally. The dashed line x = h is excluded from the domain and is a vertical asymptote. The horizontal asymptote remains y = 0.

f(-1)

-1

f(0)

Undefined

f(1)

1

For f(x) = 1/x, find f(2).

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✓1/2 = 0.5. At x = 0 there is no function value. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

The variable is in the exponent. For a > 1, the function grows; for 0 < a < 1, it decays. The domain is all real numbers, the range is positive, and the graph passes through (0, 1).

This control uses bases greater than 1. Compare the solid exponential with its dashed logarithmic inverse: they reflect across the dotted line y = x.

f(-1)

0.5

f(0)

1

f(1)

2

For f(x) = 2ˣ, find f(3).

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✓2³ = 2 × 2 × 2 = 8. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

A logarithm asks for an exponent: log₂(8) = 3 because 2³ = 8. Its domain is x > 0 and its range is all real numbers. It reverses an exponential function and passes through (1, 0).

This control uses bases greater than 1. The dashed exponential reverses the solid logarithm. They reflect across y = x; the logarithm is undefined for x ≤ 0.

f(-1)

Undefined

f(0)

Undefined

f(1)

0

For f(x) = log₂(x), find f(8).

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✓2³ = 8, so log₂(8) = 3. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

Use radians. Sine and cosine repeat every 2π, have domain ℝ and range [−1, 1]. Tangent is sin(x)/cos(x), repeats every π, and is undefined at π/2 + kπ for integer k. Its range is ℝ.

Change the amplitude of A sin(x). Its range becomes [−A, A], while its period stays 2π. The x-axis uses radians.

f(-1)

-0.841

f(0)

0

f(1)

0.841

Find cos(0).

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✓cos(0) = 1, while sin(0) = 0. Not yet. Use the function stated in the question and substitute the requested input. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.