NY Regents Algebra I: a complete guide to functions and statistics on the exam

What this strand demands

The functions-and-statistics strand carries a large share of the NY Regents Algebra I exam. It rewards reading functions (evaluate, find key features), choosing and building the right model (linear, exponential, or quadratic), and analyzing data (center and spread for one variable, regression for two). This guide ties together the dot-point pages, each with its own practice: function notation and key features, linear and exponential models, quadratic functions and their graphs, one-variable statistics, and two-variable data and regression.

Functions and their key features

A function gives exactly one output per input (the vertical-line test). Function notation $f(x)$ names the output at input $x$, so evaluating means substituting. The domain is the allowed inputs and the range is the outputs, both restricted to sensible values in a context.

From a graph, read the intercepts (the zeros and the starting value), the intervals of increase and decrease, the relative maxima and minima, and the average rate of change $\frac{f(b) - f(a)}{b - a}$, which is the slope of the line through the endpoints.

Linear and exponential models

A constant difference signals a linear model $f(x) = mx + b$; a constant ratio signals an exponential model $f(x) = a(b)^x$. The slope $m$ and initial value $b$ define a line; the initial value $a$ and growth factor $b = 1 \pm r$ define an exponential. Over the long run, an exponential model always overtakes a linear one.

Quadratic functions and their graphs

A quadratic graphs as a parabola with axis of symmetry $x = -\frac{b}{2a}$ and vertex on that line. It opens up ($a > 0$, minimum) or down ($a < 0$, maximum). The y-intercept is $c$; the zeros come from factoring or the formula. Vertex form $a(x - h)^2 + k$ shows the vertex $(h, k)$, and transformations of $y = x^2$ shift, reflect, and stretch the parent: remember $(x - h)^2$ shifts right by $h$.

One-variable statistics

The mean is the average and the median is the resistant middle. Spread is the range, the resistant interquartile range $\text{IQR} = Q_3 - Q_1$, and the standard deviation (typical distance from the mean, read from a calculator). A value is an outlier beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$. Box plots show the five-number summary; to compare two distributions, compare one center and one spread.

Two-variable data and regression

A scatter plot's form, direction, and strength describe a relationship. The line of best fit $\hat{y} = mx + b$ predicts $y$ from $x$: the slope is the predicted change per unit, the intercept the predicted value at $x = 0$. A residual is actual minus predicted (positive above the line, negative below). The correlation coefficient $r$ measures linear strength, but correlation never proves causation.

How this strand is examined

• Part I (2 credits). Evaluate a function, identify a vertex or axis, classify a model, read a slope's meaning, or compare a mean and median.
• Part II (2 credits). An average rate of change with units, an outlier check with the IQR rule, or a residual computation. Show the key step.
• Part III and IV (4 to 6 credits). A projectile-motion quadratic, an exponential growth model, or a regression task with prediction and residual interpretation. Show every step and interpret in context.

Check your knowledge

Work these as you would for credit on the exam.

1. If $f(x) = 3x^2 - x$, find $f(-2)$. (2 credits)
2. A graph passes through $(1, 5)$ and $(4, 20)$. Find the average rate of change. (2 credits)
3. A culture starts at 50 and triples each hour. Write the model. (2 credits)
4. Find the vertex of $y = x^2 - 8x + 10$. (2 credits)
5. For $5, 9, 9, 12, 20$, compare the mean and median. (2 credits)
6. A data set has $Q_1 = 15$, $Q_3 = 27$. Is 50 an outlier? Show work. (2 credits)
7. For $\hat{y} = 4x + 3$, predict $y$ at $x = 6$, then find the residual if the actual value is 25. (4 credits)
8. Interpret the slope and intercept of $\hat{y} = -1.2x + 50$ for battery percent versus hours. (4 credits)