Pre-Analysis — Reading a Graph: Basics
Pre-Analysis — Reading a Graph: Basics. Practice questions to deepen understanding of reading a graph in pre-analysis — without differentiation. Online math practice with full solutions and step-by-step explanations.
Pre-analysis reading a graph basics practice — domain, range, points of intersection with the axes, sign of the function. Understanding graphs without differentiation. Domain, range, intersection points, sign.
📊 Domain:
What is the domain of a function?
| 📊 Domain Definition: Domain = all \(x\)-values for which \(f(x)\) is defined Denoted: \(D_f\) or Domain How to read it from a graph? 1️⃣ Look at the \(x\)-axis 2️⃣ Where is there a graph? → in the domain 3️⃣ Where is there no graph? → not in the domain Example: If the graph exists from \(x=-2\) to \(x=5\) Domain: \([-2, 5]\) |
📈 Range:
What is the range of a function?
| Definition: The range is the set of all output values \(y\) the function can produce. Denoted \(R_f\) or Range. How to read from a graph: Look at the \(y\)-axis: what is the lowest point? What is the highest? Example: If the graph reaches between \(y=-3\) and \(y=7\), then Range = \([-3,7]\). |
✂️ y-intercept:
How do you find the intersection point with the \(y\)-axis?
| The rule: The \(y\)-axis consists of all points with \(x=0\). Substitute \(x=0\) → the intersection point is \((0, f(0))\). Example: \(f(x)=x^2+3\) → \(f(0)=3\) → intersection: \((0,3)\). |
✂️ x-intercept:
How do you find the intersection points with the \(x\)-axis?
| The rule: The \(x\)-axis consists of all points with \(y=0\). Solve \(f(x)=0\) → points have the form \((x_1,0),(x_2,0),\ldots\) Example: \(f(x)=x^2-4\) → \(x^2-4=0\) → \(x=\pm 2\) → points \((-2,0)\) and \((2,0)\). There can be multiple intersection points! |
➕ Function sign:
When is \(f(x) > 0\) (positive)?
| ➕ Positive function The rule: \(f(x) > 0\) ↓ The graph is above the \(x\)-axis How to read it from a graph? 1️⃣ Look at the \(x\)-axis 2️⃣ Where is the graph above the axis? 3️⃣ Those are the intervals where \(f(x) > 0\) Remember: Above the axis = positive \(y > 0\) because we are above \(y=0\) |
➖ Function sign:
When is \(f(x) < 0\) (negative)?
| ➖ Negative function The rule: \(f(x) < 0\) ↓ The graph is below the \(x\)-axis How to read it from a graph? 1️⃣ Look at the \(x\)-axis 2️⃣ Where is the graph below the axis? 3️⃣ Those are the intervals where \(f(x) < 0\) Remember: Below the axis = negative \(y < 0\) because we are below \(y=0\) |
📊 Exercise:
A graph is defined between \(x=-3\) and \(x=5\). What is the domain?
| Reading the domain Solution: The graph is defined from \(x=-3\) to \(x=5\) Domain: \([-3, 5]\) Why square brackets? \([a, b]\) = includes the endpoints \((a, b)\) = excludes the endpoints If the graph exists at \(x=-3\) and \(x=5\) → we include them! Note: Always write from smallest to largest: \([-3, 5]\) ✓ \([5, -3]\) ✗ |
📈 Exercise:
The lowest height of a graph is \(y=-2\) and the highest is \(y=4\). What is the range?
| Reading the range Solution: Lowest: \(y=-2\) Highest: \(y=4\) Range: \([-2, 4]\) How to read it? 1️⃣ Look at the \(y\)-axis 2️⃣ What is the minimum? \(-2\) 3️⃣ What is the maximum? \(4\) 4️⃣ Write: \([-2, 4]\) |
🎯 Zero points:
What are the points where \(f(x)=0\) called?
| 🎯 Zero points Different names: ✅ Zero points ✅ Roots of the function ✅ Intersection points with the \(x\)-axis They all mean the same thing! What are they? The points where \(f(x) = 0\) Graphically: where the graph crosses the \(x\)-axis Why "zero"? Because the function value = 0 \(y = f(x) = 0\) |
📚 Summary:
What do you need to know to read a basic graph?
📚 Summary - Reading a graph 1️⃣ Domain: All \(x\)-values for which the graph exists Look at the \(x\)-axis 2️⃣ Range: All \(y\)-values the function takes Look at the \(y\)-axis 3️⃣ y-intercept: Substitute \(x=0\) Point: \((0, f(0))\) 4️⃣ x-intercept: Solve \(f(x)=0\) Points: \((x_i, 0)\) 5️⃣ Sign: Above the \(x\)-axis → positive Below the \(x\)-axis → negative |