Writing Functions from Graphs – College Algebra Guidance

Q5. Write the function represented by each graph.

Background

Topic: Graphs of Functions

This question tests your ability to interpret a graph and write the corresponding algebraic function. You need to identify the type of function (quadratic, cubic, absolute value, etc.) and use key points from the graph to determine its equation.

Key Terms and Formulas:

• Quadratic Function: $y = ax^2 + bx + c$

• Cubic Function: $y = ax^3 + bx^2 + cx + d$

• Absolute Value Function: $y = a|x - h| + k$

• Vertex: The turning point of a parabola or absolute value graph.

• Intercepts: Points where the graph crosses the axes.

Step-by-Step Guidance

1. Examine the shape of each graph to determine the type of function (e.g., parabola for quadratic, V-shape for absolute value, S-shape for cubic).

2. Identify key points on the graph, such as the vertex, intercepts, or other labeled points. These will help you set up equations for the function.

3. For a quadratic function, use the vertex form $y = a(x - h)^2 + k$ or standard form $y = ax^2 + bx + c$. For absolute value, use $y = a|x - h| + k$. For cubic, use $y = ax^3 + bx^2 + cx + d$.

4. Plug the coordinates of the key points into the chosen function form to create a system of equations. Solve for the unknown coefficients (a, b, c, etc.).

5. Check your function by substituting other points from the graph to ensure the equation fits all given data.

Try solving on your own before revealing the answer!

Final Answer:

a. $f(x) = -\frac{1}{2}(x - 2)^2 - 3$ (quadratic, vertex at (2, -3))

b. $f(x) = -x^2 - 2$ (quadratic, vertex at (0, -2))

c. $f(x) = (x - 2)^3 + 2$ (cubic, inflection at (2, 2))

d. $f(x) = |x - 2| - 3$ (absolute value, vertex at (2, -3))

Each function matches the key features and points shown in the corresponding graph.