An expression is shown below:
f(x) = 2x2 − 5x + 3
Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
Part A: To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.
Setting f(x) = 0, we have:
2x^2 - 5x + 3 = 0
This is a quadratic equation that can be factored. We need to find two numbers whose product is 2 * 3 = 6 and whose sum is -5, the coefficient of x. The numbers -2 and -3 satisfy these conditions, so we can factor the equation as:
(2x - 3)(x - 1) = 0
Now we can set each factor equal to zero and solve for x:
2x - 3 = 0 or x - 1 = 0
Solving these equations, we find:
2x = 3 or x = 1
x = 3/2 or x = 1
Therefore, the x-intercepts of the graph of f(x) are x = 3/2 and x = 1.
Part B: The vertex of the graph of f(x) corresponds to the minimum or maximum point. To determine whether it is a maximum or minimum, we need to consider the coefficient of x^2 in the quadratic equation. If the coefficient is positive, the parabola opens upwards and the vertex represents the minimum point. If the coefficient is negative, the parabola opens downwards and the vertex represents the maximum point.
In the given expression, the coefficient of x^2 is positive (2), so the parabola opens upwards. Therefore, the vertex represents the minimum point.
To find the coordinates of the vertex, we can use the formula:
x-coordinate of the vertex = -b / (2a)
where a and b are the coefficients of x^2 and x, respectively. From the given expression, a = 2 and b = -5.
Substituting the values into the formula:
x-coordinate of the vertex = -(-5) / (2 * 2) = 5/4
To find the y-coordinate, we substitute the x-coordinate back into the expression:
f(x) = 2(5/4)^2 - 5(5/4) + 3
f(x) = 2(25/16) - 25/4 + 3
f(x) = 25/8 - 25/4 + 3
f(x) = 25/8 - 50/8 + 24/8
f(x) = -1/8
Therefore, the coordinates of the vertex are (5/4, -1/8).
Part C: The steps to graph f(x) are as follows:
1. Plot the x-intercepts of the graph obtained in Part A. Mark the points (3/2, 0) and (1, 0).
2. Plot the vertex obtained in Part B. Mark the point (5/4, -1/8).
3. Determine the shape of the graph. Since the coefficient of x^2 is positive, it is a U-shaped parabola opening upwards.
4. Draw the graph by connecting the plotted points and accurately shaping the parabola. The graph should pass through the vertex and the x-intercepts.
We can use the answers obtained in Part A and Part B to draw the graph because they provide us with critical points on the graph: the x-intercepts and the vertex. Plotting these points and understanding the shape of the graph helps us accurately sketch the graph of f(x).
Part A: To find the x-intercepts of the graph of f(x), we need to find the values of x when f(x) equals zero. In other words, we need to solve the equation 2x^2 - 5x + 3 = 0.
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation. In our case, a = 2, b = -5, and c = 3.
Using the quadratic formula, we can plug in these values and solve for x:
x = (-(-5) ± √((-5)^2 - 4(2)(3))) / (2(2))
x = (5 ± √(25 - 24)) / 4
x = (5 ± √1) / 4
Simplifying further, we get:
x1 = (5 + 1) / 4 = 6 / 4 = 1.5
x2 = (5 - 1) / 4 = 4 / 4 = 1
Therefore, the x-intercepts of the graph of f(x) are 1.5 and 1.
Part B: To determine whether the vertex of the graph of f(x) is a maximum or minimum, we need to look at the coefficient of x^2. If this coefficient (in front of x^2) is positive, the parabola opens upwards and the vertex is a minimum. If it is negative, the parabola opens downwards and the vertex is a maximum.
In our expression, the coefficient of x^2 is 2, which is positive. Therefore, the vertex of the graph of f(x) will be a minimum.
To find the coordinates of the vertex, we can use the formula: x = -b / (2a)
In our case, a = 2 and b = -5, so:
x = -(-5) / (2 * 2) = 5 / 4 = 1.25
To find the corresponding y-coordinate, we substitute this x-value back into the original equation:
f(1.25) = 2(1.25)^2 - 5(1.25) + 3 = 2.625 - 6.25 + 3 = -0.625
Therefore, the coordinates of the vertex (minimum) are (1.25, -0.625).
Part C: To graph f(x), we can follow these steps:
1. Plot the x-intercepts: We found the x-intercepts to be 1.5 and 1. Plot these points on the x-axis.
2. Plot the vertex: We found the vertex to be (1.25, -0.625). Plot this point on the graph.
3. Choose additional points: Choose a few more x-values, plug them into the original equation f(x), and calculate the corresponding y-values. Plot these points on the graph.
4. Draw the graph: Connect the plotted points with a smooth curve. Note that since we determined the vertex to be a minimum, the parabola should be U-shaped and open upwards.
With the information obtained from Part A and Part B, we know the x-intercepts, the vertex, and the shape of the graph, allowing us to accurately sketch the graph of f(x).