Function Graphs: Increasing and Decreasing Intervals on the Digital SAT

Quick Answer: A function is increasing when its y-values go up as x-values move to the right, and decreasing when y-values go down. On the Digital SAT, graphing the function in Desmos is the fastest way to visually identify the exact intervals between local minimums and maximums.

graph LR
    A[Enter Equation] --> B[Identify Peaks/Valleys] --> C[Find X-Coordinates] --> D[Determine Direction] --> E[Write Interval]

What Are Function Graphs: Increasing and Decreasing Intervals?

On the College Board Digital SAT, you will frequently encounter questions asking you to analyze the behavior of a function. A function is considered increasing on an interval if the $y$-values get larger as the $x$-values get larger (moving left to right). Conversely, it is decreasing if the $y$-values get smaller as the $x$-values get larger.

These intervals are always defined by their $x$-values. The points where a function switches from increasing to decreasing (or vice versa) are called turning points, which correspond to local maximums or minimums. For example, if you are looking at the vertex form of a quadratic, the vertex is the exact turning point where the parabola changes direction.

Step-by-Step Method

1. Step 1 — Identify the turning points (local maximums and minimums) of the function on the coordinate plane.
2. Step 2 — Find the $x$-coordinates of these turning points. These $x$-values act as the boundaries for your intervals.
3. Step 3 — Look at the sections of the graph between these boundaries. If the graph goes "uphill" from left to right, it's increasing. If it goes "downhill," it's decreasing.
4. Step 4 — Write the interval using the $x$-values (for example, "increasing for $x < 3$").

Desmos Shortcut

The absolute best way to handle these questions is by using the built-in Desmos Calculator. Type the function equation directly into Desmos. Click on the peaks and valleys (the maximum and minimum points) of the graph; Desmos will automatically reveal their coordinates in gray. Look entirely at the $x$-coordinate of that point to determine where the interval splits.

Worked Example

Question: The function $f$ is defined by $f(x) = -2x^2 + 12x - 10$. For what interval is the function decreasing? A) $x < 3$ B) $x > 3$ C) $x < 8$ D) $x > 8$

Solution: First, you can find the vertex of this parabola. Since it's in standard form, the $x$-coordinate of the vertex is at $x = \frac{-b}{2a}$. $x = \frac{-12}{2(-2)} = \frac{-12}{-4} = 3$

Since the leading coefficient is negative ($-2$), the parabola opens downward, meaning it has a maximum at $x = 3$. The graph goes up (increases) until it hits $x = 3$, and then goes down (decreases) after $x = 3$. Therefore, the function is decreasing for all $x$-values greater than 3.

Alternatively, typing $y = -2x^2 + 12x - 10$ into Desmos immediately shows a downward-facing parabola with a peak at $(3, 8)$. Looking to the right of $x = 3$, the graph goes down.

Correct Answer: B

Common Traps

1. Using y-values instead of x-values — When asked "where" a function is increasing or decreasing, students often mistakenly pick the $y$-value of the vertex. Intervals are always stated in terms of the domain ($x$-values).

2. Algebraic sign errors — Based on Lumist student data, 15% of errors in Advanced Math involve confusing the vertex form $a(x-h)^2+k$ by getting the $h$ sign wrong. If you convert standard form to vertex form or try factoring quadratics by hand, a simple negative sign error will shift your entire interval. Our data shows that students who graph quadratics in Desmos before solving identify vertex/roots 35% faster and avoid these algebraic traps entirely.

FAQ

How do I know if I should use brackets or parentheses for intervals?

On the SAT, increasing and decreasing intervals are typically written using strict inequalities (like $x > 2$) rather than interval notation. At the exact turning point, the function is neither increasing nor decreasing.

Does this only apply to parabolas?

No, you will see increasing and decreasing intervals on linear, exponential, and higher-degree polynomial functions. However, quadratics are the most commonly tested function type for turning points.

What if the function is a straight line?

For a linear function, the slope determines its behavior. A positive slope means it is increasing everywhere, and a negative slope means it is decreasing everywhere.

How many Function Graphs: Increasing and Decreasing Intervals questions are on the SAT?

Advanced Math makes up approximately 35% of SAT Math. On Lumist.ai, we have 20 practice questions specifically on this topic to help you master turning points and graph analysis.