Sure, let's go through this step by step to understand the function [tex]\( f(x) = -5x^2 + 6x + 1.4 \)[/tex].
### Step 1: Identify the coefficients
The function [tex]\( f(x) = -5x^2 + 6x + 1.4 \)[/tex] is a quadratic function, which generally has the form [tex]\( ax^2 + bx + c \)[/tex]. Here:
- [tex]\( a = -5 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = 1.4 \)[/tex]
### Step 2: Determine the axis of symmetry
For a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the axis of symmetry can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plug in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{6}{2(-5)} = -\frac{6}{-10} = \frac{3}{5} = 0.6 \][/tex]
So the axis of symmetry is [tex]\( x = 0.6 \)[/tex].
### Step 3: Find the vertex
The vertex of the quadratic function is on the axis of symmetry. To find the y-coordinate of the vertex, substitute [tex]\( x = 0.6 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0.6) = -5(0.6)^2 + 6(0.6) + 1.4 \][/tex]
Calculate each term:
[tex]\[ -5(0.6)^2 = -5(0.36) = -1.8 \][/tex]
[tex]\[ 6(0.6) = 3.6 \][/tex]
Now sum all terms:
[tex]\[ f(0.6) = -1.8 + 3.6 + 1.4 = 3.2 \][/tex]
So the vertex is at [tex]\( (0.6, 3.2) \)[/tex].
### Step 4: Determine the direction of the parabola
Since the coefficient [tex]\( a = -5 \)[/tex] is negative, the parabola opens downwards.
### Step 5: Finding the y-intercept
The y-intercept of the function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -5(0)^2 + 6(0) + 1.4 = 1.4 \][/tex]
So the y-intercept is [tex]\( (0, 1.4) \)[/tex].
### Step 6: Finding the x-intercepts (if any)
The x-intercepts (roots) are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. Solve the equation:
[tex]\[ -5x^2 + 6x + 1.4 = 0 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ a = -5, \; b = 6, \; c = 1.4 \][/tex]
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 6^2 - 4(-5)(1.4) = 36 + 28 = 64 \][/tex]
Since the discriminant is positive, there are two real roots:
[tex]\[ x = \frac{-6 \pm \sqrt{64}}{2(-5)} = \frac{-6 \pm 8}{-10} \][/tex]
Calculate the roots:
[tex]\[ x = \frac{-6 + 8}{-10} = \frac{2}{-10} = -0.2 \][/tex]
[tex]\[ x = \frac{-6 - 8}{-10} = \frac{-14}{-10} = 1.4 \][/tex]
So the x-intercepts are [tex]\( x = -0.2 \)[/tex] and [tex]\( x = 1.4 \)[/tex].
### Summary
- Vertex: [tex]\( (0.6, 3.2) \)[/tex]
- Axis of symmetry: [tex]\( x = 0.6 \)[/tex]
- Direction: Downward
- y-intercept: [tex]\( (0, 1.4) \)[/tex]
- x-intercepts: [tex]\( x = -0.2 \)[/tex] and [tex]\( x = 1.4 \)[/tex]
This provides a detailed description of the quadratic function [tex]\( f(x) = -5x^2 + 6x + 1.4 \)[/tex].
