Question 70·Easy·Nonlinear Functions
The function is defined by . What are the coordinates of the vertex of the graph of ?
When a quadratic is already in vertex form , go straight to identifying and and write the vertex as —this is much faster than expanding or doing any calculus. Pay close attention to signs: means , while would mean , and the constant at the end is the -coordinate of the vertex. This quick pattern recognition saves time and avoids sign mistakes.
Hints
Identify the form of the quadratic
Ask yourself: Is the function in the standard form or in the vertex form ?
Recall what vertex form tells you
In vertex form , the vertex of the parabola is at . Focus on how and appear in the expression.
Be careful with signs
For , the value of is the number that makes the expression match, so means , not . The value is just the constant added or subtracted at the end.
Desmos Guide
Graph the function
Type y = (x - 3)^2 + 4 into Desmos and press Enter to display the parabola.
Find and read the vertex
Click on the lowest point (for this upward-opening parabola) or use the "maximum/minimum" feature; Desmos will show the coordinates of this point. Those coordinates are the vertex of the graph.
Step-by-step Explanation
Recognize vertex form of a quadratic
A quadratic can be written in vertex form as
In this form, the vertex of the parabola is the point .
Match the given function to vertex form
The given function is .
Compare it to :
- (since we just have )
- matches , so
- is the constant added at the end, so .
Write the vertex coordinates
From the comparison, and , so the vertex of the graph is the point .