Linear regression — predict with a line

Background

Linear regression models a continuous output as a straight line. With a single input feature the model is

$$\hat{y} = w x + b$$

where $w$ (the weight / slope, written $m$ in the lesson) tilts the line and $b$ (the bias / intercept, written $c$) shifts it up or down. Predicting is just evaluating this line at each input — the very first thing every later step (residuals, SSR, gradients) is built on.

Problem statement

Implement predict(x, w, b) that returns the line's prediction $\hat{y} = w x + b$ for every value in x.

Input

• x — array-like of input features (any length).
• w — float, the slope.
• b — float, the intercept.

Output

Returns an np.ndarray of the same length as x, where element $i$ is $w x_i + b$.

Examples

Example 1

Input:  x = [2, 4, 5], w = 3, b = -4
Output: [2. 8. 11.]

Explanation: the lesson's line $y = 3x - 4$ gives $3\cdot2-4=2$, $3\cdot4-4=8$, $3\cdot5-4=11$.

Example 2

Input:  x = [51], w = 1, b = 1
Output: [52.]

Explanation: the lesson's starting guess $m=1, c=1$ predicts $52$ at $x=51$ — far from the true $149$, which is why training is needed.

Constraints

• Convert x to a NumPy array so the arithmetic is elementwise.
• Do not loop in Python — one vectorised expression is enough.

Notes

• This is the model's forward pass. Every other quantity in the lesson — residuals, SSR, gradients — starts from this prediction.