Grade 10 Math MCAS: a complete guide to the Functions category

What this category demands

The Functions category is one of the largest on the Grade 10 MCAS, drawn from the F-IF, F-BF, and F-LE standards of the 2017 framework. It wants you to read a function in any representation (graph, table, equation, words), to name its key features, to model with linear and exponential functions, and to describe how transformations move a graph. This guide ties together the dot-point pages, each with its own practice: function notation and key features, linear functions and rate of change, quadratic functions and their graphs, exponential functions and growth, comparing and building functions, and transformations of functions.

Function notation and key features

The notation $f(x)$ names the output for an input; evaluate by substitution. The domain is the set of inputs, the range the set of outputs. Key features read from a graph: the y-intercept at $x = 0$, the x-intercepts (zeros) where $f(x) = 0$, intervals where the function increases or decreases, and any maximum or minimum. A relation is a function only if each input has one output (the vertical line test).

Linear functions and rate of change

Slope is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, rise over run, and it is not on the reference sheet. Write lines in slope-intercept form $y = mx + b$ (slope and y-intercept visible) or point-slope form $y - y_1 = m(x - x_1)$ (from a slope and a point). In context, slope is a constant rate of change with units. Parallel lines share a slope; perpendicular slopes are negative reciprocals (product $-1$).

Quadratic functions and graphs

A quadratic graphs as a parabola, opening up if $a > 0$ (minimum) and down if $a < 0$ (maximum). The axis of symmetry is $x = -\dfrac{b}{2a}$, and the vertex is found by substituting that $x$ back in. The three forms reveal features: standard shows the y-intercept $c$, factored shows the zeros, and vertex form $a(x - h)^2 + k$ shows the vertex, with the sign of $h$ flipped.

Exponential functions and growth

An exponential function $f(x) = A_0 b^x$ has initial value $A_0$ and base $b$. Growth has $b > 1$ ($b = 1 + r$); decay has $0 < b < 1$ ($b = 1 - r$). The defining contrast with linear functions: linear changes by a constant difference, exponential by a constant ratio, and an exponential eventually outgrows any line.

Comparing and building functions

To compare two functions in different forms, read the same feature (slope, intercept, maximum) from each, then compare numbers. To build a function, decide linear (constant difference) or exponential (constant ratio) from the data, then find the parameters. The average rate of change over $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$, the slope of the line joining the endpoints.

Transformations

Transformations of $f(x)$: $f(x) + k$ shifts vertically by $k$; $f(x - h)$ shifts horizontally by $h$ (sign reversed); $-f(x)$ reflects across the x-axis; $f(-x)$ across the y-axis; $a \cdot f(x)$ stretches ($|a| > 1$) or compresses ($|a| < 1$). Vertex form $a(x - h)^2 + k$ packages all of these for a parabola.

How this category is examined

• Selected-response (1 point, mixed calculator). Evaluate a function, read an intercept or vertex, identify growth rate, match a transformation. Technology-enhanced items may ask you to plot a line or parabola.
• Short-answer (1 point). A single evaluation, a slope, an average rate of change, or one transformation described.
• Constructed-response (multi-point). Analyze a parabola fully (axis, vertex, intercepts, zeros), build and apply a model, or compare functions with justification.

Check your knowledge

Work these as you would for credit.

1. If $f(x) = 2x^2 - 3x$, find $f(-2)$. (1 point)
2. Find the slope through $(1, -2)$ and $(5, 6)$. (1 point)
3. Write the equation of the line with slope $-2$ through $(3, 1)$. (2 points)
4. Find the vertex and axis of symmetry of $y = x^2 - 8x + 12$. (2 points)
5. A culture of 200 bacteria doubles every hour. Write a function for the count after $h$ hours. (2 points)
6. A table shows $y = 5, 10, 20, 40$ for $x = 0, 1, 2, 3$. Linear or exponential, and what is the model? (2 points)
7. Find the average rate of change of $f(x) = x^2$ from $x = 2$ to $x = 5$. (2 points)
8. Describe how $g(x) = (x + 2)^2 - 3$ transforms $f(x) = x^2$. (2 points)