To find the y-intercept and the minimum of the quadratic function [tex]\( y = x^2 + 4x - 5 \)[/tex], let's go through the steps involved.
### Finding the y-intercept
The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
y = (0)^2 + 4(0) - 5 = -5
\][/tex]
So, the y-intercept of the function is [tex]\( (0, -5) \)[/tex].
### Finding the minimum of the function
A quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex] has a parabolic shape. If the coefficient [tex]\( a \)[/tex] is positive, the parabola opens upwards and the vertex represents the minimum point of the function.
The vertex (h, k) of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] can be found using:
[tex]\[ h = -\frac{b}{2a} \][/tex]
For our function [tex]\( y = x^2 + 4x - 5 \)[/tex]:
1. The coefficients are [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex].
2. Calculate [tex]\( h \)[/tex] (the x-coordinate of the vertex):
[tex]\[
h = -\frac{b}{2a} = -\frac{4}{2 \cdot 1} = -2
\][/tex]
3. Substitute [tex]\( h = -2 \)[/tex] back into the function to find the y-coordinate ([tex]\( k \)[/tex]) of the vertex:
[tex]\[
y = (-2)^2 + 4(-2) - 5 = 4 - 8 - 5 = -9
\][/tex]
So, the vertex of the function, which represents the minimum point, is [tex]\((-2, -9)\)[/tex].
### Summary
- The y-intercept of the function [tex]\( y = x^2 + 4x - 5 \)[/tex] is [tex]\((0, -5)\)[/tex].
- The minimum point of the function is [tex]\((-2, -9)\)[/tex].
