Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Sure, let's analyze the given function step-by-step:
### Problem:
We're given the function [tex]\( y = x^2 - 8x \)[/tex].
### Step-by-Step Analysis:
1. Identifying the Function Type:
- The given function [tex]\( y = x^2 - 8x \)[/tex] is a quadratic function. Quadratic functions are generally of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- Comparing, we see [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Finding the Vertex:
- The vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex].
[tex]\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
- To find the y-coordinate of the vertex, substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[ y = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
- Therefore, the vertex of the function is at [tex]\( (4, -16) \)[/tex].
3. Finding the Axis of Symmetry:
- The axis of symmetry of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
[tex]\[ x = 4 \][/tex]
4. Finding the x-intercepts:
- The x-intercepts are the points where the graph of the function crosses the x-axis.
- Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = x^2 - 8x \][/tex]
- Factor the quadratic expression:
[tex]\[ 0 = x(x - 8) \][/tex]
- Solve each factor:
[tex]\[ x = 0 \quad \text{or} \quad x - 8 = 0 \Rightarrow x = 8 \][/tex]
- Therefore, the x-intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex], which are the points [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex].
5. Finding the y-intercept:
- The y-intercept is the point where the graph of the function crosses the y-axis.
- Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = 0^2 - 8(0) = 0 \][/tex]
- Therefore, the y-intercept is at [tex]\( (0, 0) \)[/tex].
### Summary:
1. Function Type: Quadratic function [tex]\( y = x^2 - 8x \)[/tex]
2. Vertex: [tex]\( (4, -16) \)[/tex]
3. Axis of Symmetry: [tex]\( x = 4 \)[/tex]
4. x-intercepts: [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex]
5. y-intercept: [tex]\( (0, 0) \)[/tex]
The graph of the given function is a parabola that opens upwards with its vertex at [tex]\( (4, -16) \)[/tex]. The parabola intersects the x-axis at [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex], and intersects the y-axis at [tex]\( (0, 0) \)[/tex].
### Problem:
We're given the function [tex]\( y = x^2 - 8x \)[/tex].
### Step-by-Step Analysis:
1. Identifying the Function Type:
- The given function [tex]\( y = x^2 - 8x \)[/tex] is a quadratic function. Quadratic functions are generally of the form [tex]\( y = ax^2 + bx + c \)[/tex].
- Comparing, we see [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Finding the Vertex:
- The vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex].
[tex]\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
- To find the y-coordinate of the vertex, substitute [tex]\( x = 4 \)[/tex] back into the function:
[tex]\[ y = 4^2 - 8(4) = 16 - 32 = -16 \][/tex]
- Therefore, the vertex of the function is at [tex]\( (4, -16) \)[/tex].
3. Finding the Axis of Symmetry:
- The axis of symmetry of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
[tex]\[ x = 4 \][/tex]
4. Finding the x-intercepts:
- The x-intercepts are the points where the graph of the function crosses the x-axis.
- Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = x^2 - 8x \][/tex]
- Factor the quadratic expression:
[tex]\[ 0 = x(x - 8) \][/tex]
- Solve each factor:
[tex]\[ x = 0 \quad \text{or} \quad x - 8 = 0 \Rightarrow x = 8 \][/tex]
- Therefore, the x-intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex], which are the points [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex].
5. Finding the y-intercept:
- The y-intercept is the point where the graph of the function crosses the y-axis.
- Set [tex]\( x = 0 \)[/tex] in the equation:
[tex]\[ y = 0^2 - 8(0) = 0 \][/tex]
- Therefore, the y-intercept is at [tex]\( (0, 0) \)[/tex].
### Summary:
1. Function Type: Quadratic function [tex]\( y = x^2 - 8x \)[/tex]
2. Vertex: [tex]\( (4, -16) \)[/tex]
3. Axis of Symmetry: [tex]\( x = 4 \)[/tex]
4. x-intercepts: [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex]
5. y-intercept: [tex]\( (0, 0) \)[/tex]
The graph of the given function is a parabola that opens upwards with its vertex at [tex]\( (4, -16) \)[/tex]. The parabola intersects the x-axis at [tex]\( (0, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex], and intersects the y-axis at [tex]\( (0, 0) \)[/tex].
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
