IDNLearn.com: Your trusted source for finding accurate answers. Find in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
Sure, let's go through each part of the question step-by-step.
### 1. Finding the y-intercept
The y-intercept of a function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
Plugging [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = -5x^2 + 10x - 3 \)[/tex]:
[tex]\[ f(0) = -5(0)^2 + 10(0) - 3 = -3 \][/tex]
So, the y-intercept is:
[tex]\[ \boxed{-3} \][/tex]
### 2. Finding the x-intercepts
The x-intercepts occur where the function [tex]\( f(x) = 0 \)[/tex]. In other words, we need to solve the quadratic equation:
[tex]\[ -5x^2 + 10x - 3 = 0 \][/tex]
This is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = -3 \)[/tex].
The solutions to this quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 10^2 - 4(-5)(-3) = 100 - 60 = 40 \][/tex]
Next, we take the square root of the discriminant:
[tex]\[ \sqrt{40} \approx 6.3246 \][/tex]
Now, substitute into the quadratic formula:
[tex]\[ x = \frac{-10 \pm 6.3246}{-10} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{10 + 6.3246}{10} = 1.6325 \][/tex]
[tex]\[ x_2 = \frac{10 - 6.3246}{10} = 0.3675 \][/tex]
So, the x-intercepts are approximately:
[tex]\[ \boxed{0.3675 \text{ and } 1.6325} \][/tex]
### 3. Finding the Turning Point (Vertex)
The turning point of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] is at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -5 \)[/tex] and [tex]\( b = 10 \)[/tex]:
[tex]\[ x = -\frac{10}{2(-5)} = 1 \][/tex]
Now we need to find [tex]\( f(1) \)[/tex] to get the y-coordinate of the vertex:
[tex]\[ f(1) = -5(1)^2 + 10(1) - 3 = -5 + 10 - 3 = 2 \][/tex]
So, the turning point (vertex) is at:
[tex]\[ \boxed{(1, 2)} \][/tex]
Summary:
- The y-intercept is [tex]\( -3 \)[/tex].
- The x-intercepts are approximately [tex]\( 0.3675 \)[/tex] and [tex]\( 1.6325 \)[/tex].
- The turning point is at [tex]\( (1, 2) \)[/tex].
### 1. Finding the y-intercept
The y-intercept of a function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
Plugging [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = -5x^2 + 10x - 3 \)[/tex]:
[tex]\[ f(0) = -5(0)^2 + 10(0) - 3 = -3 \][/tex]
So, the y-intercept is:
[tex]\[ \boxed{-3} \][/tex]
### 2. Finding the x-intercepts
The x-intercepts occur where the function [tex]\( f(x) = 0 \)[/tex]. In other words, we need to solve the quadratic equation:
[tex]\[ -5x^2 + 10x - 3 = 0 \][/tex]
This is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = -3 \)[/tex].
The solutions to this quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 10^2 - 4(-5)(-3) = 100 - 60 = 40 \][/tex]
Next, we take the square root of the discriminant:
[tex]\[ \sqrt{40} \approx 6.3246 \][/tex]
Now, substitute into the quadratic formula:
[tex]\[ x = \frac{-10 \pm 6.3246}{-10} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{10 + 6.3246}{10} = 1.6325 \][/tex]
[tex]\[ x_2 = \frac{10 - 6.3246}{10} = 0.3675 \][/tex]
So, the x-intercepts are approximately:
[tex]\[ \boxed{0.3675 \text{ and } 1.6325} \][/tex]
### 3. Finding the Turning Point (Vertex)
The turning point of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] is at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -5 \)[/tex] and [tex]\( b = 10 \)[/tex]:
[tex]\[ x = -\frac{10}{2(-5)} = 1 \][/tex]
Now we need to find [tex]\( f(1) \)[/tex] to get the y-coordinate of the vertex:
[tex]\[ f(1) = -5(1)^2 + 10(1) - 3 = -5 + 10 - 3 = 2 \][/tex]
So, the turning point (vertex) is at:
[tex]\[ \boxed{(1, 2)} \][/tex]
Summary:
- The y-intercept is [tex]\( -3 \)[/tex].
- The x-intercepts are approximately [tex]\( 0.3675 \)[/tex] and [tex]\( 1.6325 \)[/tex].
- The turning point is at [tex]\( (1, 2) \)[/tex].
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
