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Sagot :
Certainly! Let's solve the problem step-by-step.
### Part A: Finding the [tex]\( x \)[/tex]-intercepts of the graph of [tex]\( f(x) \)[/tex]
To find the [tex]\( x \)[/tex]-intercepts of the graph of the function [tex]\( f(x) = 2x^2 - x - 10 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. That is:
[tex]\[ 2x^2 - x - 10 = 0 \][/tex]
We can solve this quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -10 \)[/tex].
Plugging in the values:
[tex]\[ a = 2, \quad b = -1, \quad c = -10 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac = (-1)^2 - 4(2)(-10) = 1 + 80 = 81 \][/tex]
The [tex]\( x \)[/tex]-intercepts are then:
[tex]\[ x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 2} = \frac{1 \pm 9}{4} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{1 + 9}{4} = \frac{10}{4} = 2.5 \][/tex]
[tex]\[ x_2 = \frac{1 - 9}{4} = \frac{-8}{4} = -2.0 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2.5 \)[/tex] and [tex]\( x = -2.0 \)[/tex].
### Part B: Vertex of the graph
To determine if the vertex of the graph of [tex]\( f(x) \)[/tex] is a maximum or a minimum, and to find its coordinates, we use the vertex formula for a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex]. The [tex]\( x \)[/tex]-coordinate of the vertex is given by:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
Substituting the values:
[tex]\[ x_{\text{vertex}} = -\frac{-1}{2 \cdot 2} = \frac{1}{4} = 0.25 \][/tex]
Now, substituting [tex]\( x_{\text{vertex}} \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y_{\text{vertex}} = f\left(0.25\right) = 2(0.25)^2 - 0.25 - 10 \][/tex]
Calculating step-by-step:
[tex]\[ (0.25)^2 = 0.0625 \][/tex]
[tex]\[ 2 \cdot 0.0625 = 0.125 \][/tex]
[tex]\[ 0.125 - 0.25 = -0.125 \][/tex]
[tex]\[ -0.125 - 10 = -10.125 \][/tex]
So, the coordinates of the vertex are [tex]\( \left(0.25, -10.125\right) \)[/tex].
Because the coefficient [tex]\( a \)[/tex] (which is 2) is positive, the parabola opens upwards. Therefore, the vertex is a minimum point.
### Part C: Steps to graph [tex]\( f(x) \)[/tex]
Here are the steps to graph the function [tex]\( f(x) = 2x^2 - x - 10 \)[/tex]:
1. Identify the [tex]\( x \)[/tex]-intercepts:
From Part A, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2.5 \)[/tex] and [tex]\( x = -2.0 \)[/tex]. Plot these points on the [tex]\( x \)[/tex]-axis.
2. Identify the vertex:
From Part B, the vertex is at [tex]\( (0.25, -10.125) \)[/tex]. Plot this point.
3. Determine the direction in which the parabola opens:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is 2) is positive, the parabola opens upwards.
4. Draw the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. For this function, it is [tex]\( x = 0.25 \)[/tex].
5. Plot additional points, if necessary, for accuracy:
Calculate and plot additional points on either side of the vertex, if needed, to help in drawing the curve more accurately.
6. Draw the parabola:
Draw a smooth curve starting from one [tex]\( x \)[/tex]-intercept, moving through the vertex, and reaching the other [tex]\( x \)[/tex]-intercept.
By following these steps and using the [tex]\( x \)[/tex]-intercepts and the vertex, you can accurately graph the function [tex]\( f(x) \)[/tex].
### Part A: Finding the [tex]\( x \)[/tex]-intercepts of the graph of [tex]\( f(x) \)[/tex]
To find the [tex]\( x \)[/tex]-intercepts of the graph of the function [tex]\( f(x) = 2x^2 - x - 10 \)[/tex], we need to solve the equation [tex]\( f(x) = 0 \)[/tex]. That is:
[tex]\[ 2x^2 - x - 10 = 0 \][/tex]
We can solve this quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -10 \)[/tex].
Plugging in the values:
[tex]\[ a = 2, \quad b = -1, \quad c = -10 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac = (-1)^2 - 4(2)(-10) = 1 + 80 = 81 \][/tex]
The [tex]\( x \)[/tex]-intercepts are then:
[tex]\[ x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 2} = \frac{1 \pm 9}{4} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{1 + 9}{4} = \frac{10}{4} = 2.5 \][/tex]
[tex]\[ x_2 = \frac{1 - 9}{4} = \frac{-8}{4} = -2.0 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2.5 \)[/tex] and [tex]\( x = -2.0 \)[/tex].
### Part B: Vertex of the graph
To determine if the vertex of the graph of [tex]\( f(x) \)[/tex] is a maximum or a minimum, and to find its coordinates, we use the vertex formula for a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex]. The [tex]\( x \)[/tex]-coordinate of the vertex is given by:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
Substituting the values:
[tex]\[ x_{\text{vertex}} = -\frac{-1}{2 \cdot 2} = \frac{1}{4} = 0.25 \][/tex]
Now, substituting [tex]\( x_{\text{vertex}} \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y_{\text{vertex}} = f\left(0.25\right) = 2(0.25)^2 - 0.25 - 10 \][/tex]
Calculating step-by-step:
[tex]\[ (0.25)^2 = 0.0625 \][/tex]
[tex]\[ 2 \cdot 0.0625 = 0.125 \][/tex]
[tex]\[ 0.125 - 0.25 = -0.125 \][/tex]
[tex]\[ -0.125 - 10 = -10.125 \][/tex]
So, the coordinates of the vertex are [tex]\( \left(0.25, -10.125\right) \)[/tex].
Because the coefficient [tex]\( a \)[/tex] (which is 2) is positive, the parabola opens upwards. Therefore, the vertex is a minimum point.
### Part C: Steps to graph [tex]\( f(x) \)[/tex]
Here are the steps to graph the function [tex]\( f(x) = 2x^2 - x - 10 \)[/tex]:
1. Identify the [tex]\( x \)[/tex]-intercepts:
From Part A, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = 2.5 \)[/tex] and [tex]\( x = -2.0 \)[/tex]. Plot these points on the [tex]\( x \)[/tex]-axis.
2. Identify the vertex:
From Part B, the vertex is at [tex]\( (0.25, -10.125) \)[/tex]. Plot this point.
3. Determine the direction in which the parabola opens:
Since the coefficient of [tex]\( x^2 \)[/tex] (which is 2) is positive, the parabola opens upwards.
4. Draw the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. For this function, it is [tex]\( x = 0.25 \)[/tex].
5. Plot additional points, if necessary, for accuracy:
Calculate and plot additional points on either side of the vertex, if needed, to help in drawing the curve more accurately.
6. Draw the parabola:
Draw a smooth curve starting from one [tex]\( x \)[/tex]-intercept, moving through the vertex, and reaching the other [tex]\( x \)[/tex]-intercept.
By following these steps and using the [tex]\( x \)[/tex]-intercepts and the vertex, you can accurately graph the function [tex]\( f(x) \)[/tex].
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