Power and exponential graphs

A power function keeps its exponent fixed

For y = axⁿ, the coefficient a is fixed and the input x is raised to the fixed power n. For positive a, y = ax is a line and y = ax² a parabola. y = ax³ changes sign with x and passes through the origin.

For y = 2x⁻¹ = 2/x, exclude x = 0. It gives y = 2,1,1/2 at x = 1,2,4 and negative outputs at the corresponding negative inputs. Its axes are asymptotes: the curve approaches them but never touches them.

For y = x⁻² = 1/x², again exclude zero; outputs are positive on both sides and the graph is symmetric about the y-axis. Treat x⁰ as 1 for x ≠ 0 in this index-law model; the expression 0⁰ is not silently assigned a value.

Exponential growth puts the input in the exponent

For y = 2^x, each increase of 1 in x multiplies the output by 2.

The two functions agree at some inputs but have different structures. 2^x remains positive and crosses the y-axis at 1; x² reaches zero and is symmetric about the y-axis. For y = k a^x with positive integer a > 1 and positive k, the horizontal asymptote is y = 0. When a = 1, the function is constant.

Sums change features

For y = x + 2/x, keep x ≠ 0 even though the first term is defined at zero. At x = 1, the value is 3; at x = 2, it is also 3. Matching outputs do not make the function constant. Find enough points on both sides of any excluded input and never draw a joining stroke across an undefined point.

Check your understanding 1 The input is in a different place

Compare 3x² and 3 × 2^x at x = 0, 1 and 2. Does agreement at x = 2 prove equality?

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The first gives 0, 3, 12; the second gives 3, 6, 12. Agreement at one input does not prove equal functions. The first uses a fixed power of x; the second has x in the exponent.

A growth model needs a valid interval

A stated model starts with 40 cells and doubles each hour: N = 40 × 2^t. It predicts 320 cells after 3 h. It does not establish unlimited real growth; space, resources and the model’s time interval matter.