Get expert insights and reliable answers to your questions on IDNLearn.com. Our platform offers reliable and comprehensive answers to help you make informed decisions quickly and easily.
Sagot :
To determine the graph of the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex], we need to follow these steps:
1. Identify the type of function: The function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex]. Quadratic functions graph as parabolas.
2. Determine the direction of the parabola: Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a \)[/tex]) is positive ([tex]\( a = 1 \)[/tex]), the parabola opens upwards.
3. Find the vertex: The vertex of a parabola in the form [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{3}{2 \cdot 1} = -\frac{3}{2} = -1.5 \][/tex]
To find the y-coordinate of the vertex, plug [tex]\( x = -1.5 \)[/tex] back into the function:
[tex]\[ f(-1.5) = (-1.5)^2 + 3(-1.5) + 2 = 2.25 - 4.5 + 2 = -0.25 \][/tex]
Thus, the vertex is at [tex]\( (-1.5, -0.25) \)[/tex].
4. Find the y-intercept: The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 0^2 + 3(0) + 2 = 2 \][/tex]
Thus, the y-intercept is at [tex]\( (0, 2) \)[/tex].
5. Find the x-intercepts: The x-intercepts occur where [tex]\( f(x) = 0 \)[/tex]. Solve the quadratic equation:
[tex]\[ x^2 + 3x + 2 = 0 \][/tex]
Factor the quadratic:
[tex]\[ (x + 1)(x + 2) = 0 \][/tex]
Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \][/tex]
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
Thus, the x-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex].
6. Sketch or describe the graph: Based on the information:
- The parabola opens upwards.
- The vertex is at [tex]\( (-1.5, -0.25) \)[/tex].
- The y-intercept is at [tex]\( (0, 2) \)[/tex].
- The x-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex].
The graph of [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] is a parabola that opens upwards, with its vertex at [tex]\( (-1.5, -0.25) \)[/tex]. It crosses the x-axis at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex], and it crosses the y-axis at [tex]\( (0, 2) \)[/tex]. The basic shape is a symmetric curve about the vertical line passing through the vertex.
This is how you can arrive at determining the correct graph for the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex].
1. Identify the type of function: The function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex]. Quadratic functions graph as parabolas.
2. Determine the direction of the parabola: Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a \)[/tex]) is positive ([tex]\( a = 1 \)[/tex]), the parabola opens upwards.
3. Find the vertex: The vertex of a parabola in the form [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{3}{2 \cdot 1} = -\frac{3}{2} = -1.5 \][/tex]
To find the y-coordinate of the vertex, plug [tex]\( x = -1.5 \)[/tex] back into the function:
[tex]\[ f(-1.5) = (-1.5)^2 + 3(-1.5) + 2 = 2.25 - 4.5 + 2 = -0.25 \][/tex]
Thus, the vertex is at [tex]\( (-1.5, -0.25) \)[/tex].
4. Find the y-intercept: The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Plugging [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 0^2 + 3(0) + 2 = 2 \][/tex]
Thus, the y-intercept is at [tex]\( (0, 2) \)[/tex].
5. Find the x-intercepts: The x-intercepts occur where [tex]\( f(x) = 0 \)[/tex]. Solve the quadratic equation:
[tex]\[ x^2 + 3x + 2 = 0 \][/tex]
Factor the quadratic:
[tex]\[ (x + 1)(x + 2) = 0 \][/tex]
Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \][/tex]
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
Thus, the x-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex].
6. Sketch or describe the graph: Based on the information:
- The parabola opens upwards.
- The vertex is at [tex]\( (-1.5, -0.25) \)[/tex].
- The y-intercept is at [tex]\( (0, 2) \)[/tex].
- The x-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex].
The graph of [tex]\( f(x) = x^2 + 3x + 2 \)[/tex] is a parabola that opens upwards, with its vertex at [tex]\( (-1.5, -0.25) \)[/tex]. It crosses the x-axis at [tex]\( (-1, 0) \)[/tex] and [tex]\( (-2, 0) \)[/tex], and it crosses the y-axis at [tex]\( (0, 2) \)[/tex]. The basic shape is a symmetric curve about the vertical line passing through the vertex.
This is how you can arrive at determining the correct graph for the function [tex]\( f(x) = x^2 + 3x + 2 \)[/tex].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.
