Identify and interpret key features of a graph, including intercepts, intervals of increase and decrease, relative maximum and minimum, and end behavior, in context (Ohio F-IF.4, F-IF.5).

What this topic is asking

Ohio standard F-IF.4 asks you to read and interpret the key features of a function from its graph: intercepts, intervals of increase and decrease, relative maximum and minimum, positive and negative regions, and end behavior, and to explain what each means in context. This is a high-frequency skill in the Functions category, on both parts of the test.

Intercepts: where the graph crosses the axes

The intercepts are usually the first features to read.

In a context, a zero often marks "when does the quantity reach zero?" (a ball hits the ground), and the $y$-intercept marks the starting value.

Increasing, decreasing, and turning points

Moving left to right, a graph increases where it rises and decreases where it falls; it turns at a maximum or minimum.

Positive and negative regions, and end behavior

The graph is positive where it lies above the $x$-axis (output $> 0$) and negative where it lies below (output $< 0$); the zeros are the boundaries. End behavior describes what the outputs do as $x$ grows large in each direction, useful for distinguishing function families.

How Ohio examines this topic

• Multiple choice and multiple-select. Pick the increasing interval, the maximum, the zeros, or the interval where the function is positive.
• Numeric and equation response. State a zero, the $y$-intercept, or the coordinates of the maximum.
• Graphing and drag and drop. Mark a feature on the grid, or match features to descriptions.

Why features are described in terms of x

Increasing, decreasing, positive, and negative are all statements about inputs, the $x$-values, because they describe what is happening as you move along the horizontal axis. "Increasing on $x > 2$" means: as the input grows past $2$, the output grows too. A frequent error is to answer with a $y$-value (like "increasing for $y > -4$"), which describes the wrong axis. Anchoring every interval to $x$ keeps the reading consistent and matches how the test phrases the options. The output only enters when you name the value of a maximum or minimum, the highest or lowest $y$.

Why intercepts carry the most meaning in context

Of all the features, the intercepts usually carry the clearest real-world meaning, which is why items lean on them. The $y$-intercept is the output when the input is zero, almost always a starting amount: the initial height, the up-front fee, the balance at time zero. The $x$-intercepts (zeros) are where the quantity reaches zero: when a projectile lands, when a debt is paid off, when a population dies out. Reading a graph well means not just locating these points but saying what they represent in the story the function tells, which is exactly the interpretation F-IF.4 rewards.

Try this

Q1. A graph is below the $x$-axis for $-1 < x < 3$. On what interval is the function negative? [1 point]

• Cue. Negative means below the axis, so $-1 < x < 3$.

Q2. An upward parabola has vertex $(0, -9)$. What is its minimum value? [1 point]

• Cue. The vertex is the lowest point, so the minimum output is $-9$.