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Date: _________________ Finding the Vertex
Finding the Vertex
Vocabulary and basic facts
The graph of any quadratic function f ( x ) = ax^2 + bx + c is called a parabola. The graph will have one of two shapes, and the a value tells which shape it will be.
graph shape if a is positive graph shape if a is negative
Every parabola has a special point called the vertex. It’s the lowest or highest point. Every parabola is symmetric across a vertical line called the axis of symmetry. The vertex is always on this line. The line’s equation is x = [the x -coordinate of the vertex].
Because of symmetry, when a quadratic function has two zeros, the vertex and the axis of symmetry are midway between the zeros. (In the example, the zeros are x = 3 and x = 7; the vertex and axis are at x = 5.)
Three ways to find the vertex Here are three methods for finding the coordinates of the vertex, each covered by a part of today’s assignment:
(1) Calculator commands: [2nd]TRACE minimum or [2nd][TRACE]maximum.
(2) x -coordinate of the vertex is the average of the zeros.
(3) formula for x -coordinate of the vertex: x = a
b 2
Date: _________________ Finding the Vertex
Method 1: Finding the vertex on the calculator
First graph the function on the calculator ([Y=], enter formula, [GRAPH]). If you can’t see a or shape, or the vertex isn’t on screen, press [WINDOW] and adjust. Look at graph to see whether the vertex is the maximum (highest) or minimum (lowest) point. Press [2nd][TRACE]maximum or [2nd][TRACE]minimum, whichever applies. Move the cursor to the left of the vertex, then press [ENTER]. Move the cursor to the right of the vertex, then press [ENTER]. Press [ENTER] one last time, then the calculator displays the coordinates of the vertex.
You try it
1. For each function, use your calculator to decide whether the vertex is a maximum or a minimum, find the coordinates of the vertex, and write an equation for the axis of symmetry. a. f ( x ) = 2 x^2 + 4 x + 7 maximum or minimum? vertex: axis of symmetry:
b. f ( x ) = – 3 x^2 + 6 x maximum or minimum? vertex: axis of symmetry:
c. f ( x ) = – x^2 + 4 x + 10 maximum or minimum? vertex: axis of symmetry:
2. For the quadratic function f ( x ) = x^2 + 3 x – 24, make the table and graph on your calculator. Then, use the calculator to find the zeros, vertex, and y -intercept. zeros :
vertex :
y - intercept :
Date: _________________ Finding the Vertex
4. Find the zeros of these functions using factoring, then find the coordinates of the vertex.
a. f ( x ) = 5 x^2 + 20 x + 15
b. f ( x ) = 3 x^2 6 x
c. f ( x ) = 4 x^2 – 9
5. Check your answers to all parts of problems 4 and 5 by finding the vertex of each function on your calculator. Record each vertex from the calculator here, and confirm agreement with your previous answers. Fix any mistakes that you find.
4a. 5a.
4b. 5b.
4c. 5c.
Date: _________________ Finding the Vertex
Method 3: Finding the vertex using formula
a
b x 2
Here is how to find a quadratic’s vertex using a formula.
The x - coordinate of a parabola’s vertex is always x = a
b 2
Then, you can evaluate f ( x ) to find out the y -coordinate of the vertex.
Example: Find the vertex and the axis of symmetry of f ( x ) = – 3 x^2 + 12 x + 4.
Solution: x = a
b 2
y = f (2) = – 3 · 2^2 + 12 · 2 + 4 = – 12 + 24 + 4 = 16.
Answer: The vertex is (2, 16). The axis of symmetry is the line x = 2.
You try it
6. Using the formula shown above, find the vertex and the axis of symmetry for each of these functions. a. f ( x ) = 5 x^2 – 20 x + 15
b. f ( x ) = 3 x^2 + 8 x + 6
c. f ( x ) = x^2 – 7
