How to Graph a Function in 3 Easy Steps — Mashup Math - Free Printable

Actually, the image you uploaded does not contain a problem to solve — it’s an educational infographic titled:

> “How to Graph a Function in 3 Steps”

It visually presents four common types of functions (Linear, Quadratic, Rational, Logarithmic) with their graphs and equations. It’s meant to be a reference or teaching tool, not a problem with a solution required.

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✔ What the Image Shows:

#### 1. Linear Function
- Equation: `y = 2x - 3`
- Graph: A straight line with slope 2 and y-intercept at (0, -3)
- Key features: Constant rate of change

#### 2. Quadratic Function
- Equation: `y = (1/2)x² - 4`
- Graph: A parabola opening upward, vertex at (0, -4)
- Key features: Symmetric, U-shaped curve

#### 3. Rational Function
- Equation: `y = (x + 3)/(x - 2)`
- Graph: Hyperbola with vertical asymptote at x = 2 and horizontal asymptote at y = 1
- Key features: Discontinuity (asymptotes), two branches

#### 4. Logarithmic Function
- Equation: `y = log x` (assumed base 10 unless specified)
- Graph: Curve passing through (1, 0), increasing slowly, vertical asymptote at x = 0
- Key features: Only defined for x > 0

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❓ So… What is the “Problem”?

Since no explicit question or task was given in the image, here are possible interpretations and how to respond:

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## 🎯 If You Meant: “Explain How to Graph Each Function in 3 Steps”

Here’s a general 3-step method for graphing any function, applied to each type:

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🔹 Step 1: Identify Key Features

- Domain & Range
- Intercepts (x and y)
- Asymptotes (if any)
- Vertex or turning points
- Symmetry

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🔹 Step 2: Plot Key Points

- Plug in x-values to find corresponding y-values
- Include intercepts, vertex, points near asymptotes
- Use symmetry if applicable

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🔹 Step 3: Sketch the Curve

- Connect points smoothly
- Respect asymptotes (don’t cross them!)
- Show end behavior (where the graph goes as x → ±∞)

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📊 Applied to Each Function:

#### 1. Linear: `y = 2x - 3`
- Step 1: Slope = 2, y-int = -3, domain = all real numbers
- Step 2: Plot (0, -3), (1, -1), (2, 1)
- Step 3: Draw straight line through points

#### 2. Quadratic: `y = (1/2)x² - 4`
- Step 1: Vertex (0, -4), opens up, symmetric about y-axis
- Step 2: Plot vertex, (2, -2), (-2, -2), (4, 4)
- Step 3: Draw smooth U-shape through points

#### 3. Rational: `y = (x+3)/(x-2)`
- Step 1: Vertical asymptote x=2, horizontal asymptote y=1, x-int: x=-3, y-int: y=-1.5
- Step 2: Plot (-3, 0), (0, -1.5), (3, 6), (1, -4)
- Step 3: Sketch two curves approaching asymptotes

#### 4. Logarithmic: `y = log x`
- Step 1: Domain x > 0, vertical asymptote x=0, passes through (1, 0), (10, 1)
- Step 2: Plot (1, 0), (10, 1), (0.1, -1)
- Step 3: Draw curve rising slowly to the right, approaching x=0 from right

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## ✍️ Final Answer (Summary):

> The image is an educational guide showing how to graph four common function types. There is no “problem” to solve — it’s a visual reference. However, if your goal is to learn how to graph functions, follow this 3-step process:
> 1. Identify key features (intercepts, asymptotes, vertex, etc.)
> 2. Plot key points using the function equation
> 3. Sketch the curve connecting points while respecting shape and restrictions.

Let me know if you’d like to practice graphing one of these functions step-by-step — I’m happy to walk you through it!