To find the equation of the axis of symmetry and the coordinates of the vertex of the function \( y = 4x^2 + 5x - 1 \), we can use the properties of a quadratic function in the form \( y = ax^2 + bx + c \).
### Axis of Symmetry
The axis of symmetry for a quadratic function \( y = ax^2 + bx + c \) is given by the formula:
\[ x = -\frac{b}{2a} \]
In this function:
- \( a = 4 \)
- \( b = 5 \)
Substitute these values into the formula:
\[ x = -\frac{5}{2 \cdot 4} = -\frac{5}{8} \]
So, the equation of the axis of symmetry is:
\[ x = -\frac{5}{8} \]
### Vertex
The coordinates of the vertex can be found using the axis of symmetry. The x-coordinate of the vertex is \( x = -\frac{5}{8} \). To find the y-coordinate, substitute \( x = -\frac{5}{8} \) back into the original equation:
\[ y = 4\left(-\frac{5}{8}\right)^2 + 5\left(-\frac{5}{8}\right) - 1 \]
First calculate the squared term:
\[ \left(-\frac{5}{8}\right)^2 = \frac{25}{64} \]
Then, proceed to calculate the value of \( y \):
\[ y = 4 \cdot \frac{25}{64} + 5 \cdot \left(-\frac{5}{8}\right) - 1 \]
\[ y = \frac{100}{64} + \left(-\frac{25}{8}\right) - 1 \]
\[ y = \frac{100}{64} + \left(-\frac{200}{64}\right) - \frac{64}{64} \]
\[ y = \frac{100 - 200 - 64}{64} \]
\[ y = \frac{-164}{64} \]
\[ y = -\frac{41}{16} \]
So, the y-coordinate of the vertex is \( y = -\frac{41}{16} \).
### Summary
- The equation of the axis of symmetry is \( x = -\frac{5}{8} \).
- The coordinates of the vertex are \( \left(-\frac{5}{8}, -\frac{41}{16}\right) \).
10.
Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function y = 4x2 + 5x – 1.
; vertex: mc002-8.jpg
; vertex: mc002-2.jpg
; vertex: mc002-4.jpg
; vertex: mc002-6.jpg
1 answer