Identify and interpret key features of a function graph, including x- and y-intercepts, zeros, maximum or minimum values, and intervals where the function increases or decreases (A.F.1).

What this topic is asking

This part of A.F.1 asks you to identify and interpret key features of a function graph: $x$- and $y$-intercepts, zeros, the maximum or minimum, and intervals of increase and decrease. On the Virginia Algebra I SOL these are Functions items: read a feature from a graph or equation, or interpret it in context. They appear as multiple choice, fill-in-the-blank, and hot-spot graph items.

Intercepts and zeros

Two features come from the axes:

• $x$-intercepts (zeros). Points where the graph meets the $x$-axis, so $y = 0$. To find them from an equation, set $f(x) = 0$ and solve. They are also called zeros or roots, and for a quadratic they are the factoring solutions.
• $y$-intercept. The point where the graph meets the $y$-axis, so $x = 0$. Find it by evaluating $f(0)$, which for $y = mx + b$ or $y = ax^2 + bx + c$ is just the constant term $b$ or $c$.

The words matter on the SOL: "zero" and "$x$-intercept" mean the same thing, and both differ from the $y$-intercept.

Maximum and minimum

A maximum is the largest output the function reaches; a minimum is the smallest. For a parabola, this extreme value occurs at the vertex:

• A parabola opening up has a minimum at its vertex (lowest point).
• A parabola opening down has a maximum at its vertex (highest point).

The vertex's $y$-coordinate is the max or min value, and its $x$-coordinate is where that value occurs.

Intervals of increase and decrease

Reading left to right, a function is:

• Increasing where the graph goes up (output grows as input grows).
• Decreasing where the graph goes down (output shrinks as input grows).

A line is increasing everywhere (positive slope) or decreasing everywhere (negative slope). A parabola changes direction at its vertex: it decreases on one side and increases on the other.

Interpreting features in context

SOL items attach meaning to features. For a height-versus-time parabola, the maximum is the greatest height and when it occurs; a zero is when the object hits the ground (height $0$); the $y$-intercept is the starting height (at time $0$). Naming the feature and then translating it, "the zero at $t = 4$ means it lands after $4$ seconds," is the skill these items reward.

Why zeros, intercepts, and the vertex are the graph's landmarks

These features are emphasized because together they let you sketch and interpret a graph without plotting every point. The $x$-intercepts pin where the curve meets the axis, the $y$-intercept fixes where it starts, and the vertex locates the turn and the extreme value, so a parabola is essentially determined by them. They are also the features that carry meaning in applications: a zero is when a quantity runs out or an object lands, the $y$-intercept is an initial amount, and a maximum or minimum is a best or worst case (greatest profit, least cost, highest point). This is why the SOL keeps returning to them across linear, quadratic, and exponential functions: identifying the landmarks is how you read a function's behavior and its story at a glance, and it connects directly to solving (zeros) and to the vertex form of a quadratic.

How the SOL examines this topic

• Multiple choice. Identify a zero, intercept, maximum, minimum, or interval of increase or decrease.
• Fill-in-the-blank. Type the zeros or the maximum or minimum value.
• Hot spot / graph. Click the intercepts, the vertex, or the increasing portion of a graph.

Try this

Q1. What is the $y$-intercept of $f(x) = 2x^2 + 3x - 5$? [1 point]

• Cue. $f(0) = -5$, so $(0, -5)$.

Q2. An upward parabola has vertex $(2, -1)$. Is that a max or a min? [1 point]

• Cue. A minimum (lowest point), value $-1$.