f(x) = 4x^2 − 7x − 15
Part A: What are the x-intercepts of the graph of f(x)? Show your work.
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.
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.
A. f(x) = (4x+5)(x-3)
B. Since +4 > 0, the graph opens up
as always, the vertex is at x = -b/2a = 7/8
C. plot the vertex at (7/8, -18 1/16)
pick a point or two, say at x=0,1,2 and plot them. then draw a curve through them.
what are the steps to part a
To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.
Part A:
Given: f(x) = 4x^2 - 7x - 15
To find the x-intercepts, we set f(x) = 0:
0 = 4x^2 - 7x - 15
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = -7, and c = -15. Plugging these values into the quadratic formula, we have:
x = (-(-7) ± √((-7)^2 - 4(4)(-15))) / (2(4))
x = (7 ± √(49 + 240)) / 8
x = (7 ± √289) / 8
x = (7 ± 17) / 8
Thus, we have two possible solutions for x:
x1 = (7 + 17) / 8 = 24 / 8 = 3
x2 = (7 - 17) / 8 = -10 / 8 = -5/4
Therefore, the x-intercepts of the graph of f(x) are x = 3 and x = -5/4.
Now let's move on to Part B:
To determine whether the vertex of the graph of f(x) will be a maximum or minimum, we look at the coefficient of the x^2 term in the quadratic equation.
The coefficient of x^2 is positive (4), which means the parabola opens upward and the vertex represents a minimum point.
To find the coordinates of the vertex, we use the formula:
x = -b / (2a)
Given a = 4 and b = -7, we can substitute these values into the equation to find the x-coordinate:
x = -(-7) / (2 * 4) = 7 / 8
To find the y-coordinate, we substitute this x-coordinate back into the original equation:
f(x) = 4x^2 - 7x - 15
f(7/8) = 4(7/8)^2 - 7(7/8) - 15
f(7/8) = 4(49/64) - 49/8 - 15
f(7/8) = 49/16 - 98/16 - 240/16
f(7/8) = -289/16
Therefore, the coordinates of the vertex are (7/8, -289/16).
Moving on to Part C:
To graph f(x), we can use the x-intercepts and the vertex that we found in Part A and Part B.
The x-intercepts are x = 3 and x = -5/4. These are the points where the graph intersects the x-axis.
The vertex is located at (7/8, -289/16). This is the lowest point on the graph since the parabola opens upward.
Based on this information, we can plot these points on a graph and then sketch the rest of the parabola using the symmetry of the graph. We can draw a smooth curve passing through the x-intercepts and the vertex.
So, the steps to graph f(x) would involve plotting the x-intercepts and the vertex, drawing the parabolic curve, and extending the graph in both directions.
Part A: To find the x-intercepts of the graph of f(x), we need to find the values of x where f(x) equals zero. In other words, we want to find the x-values that make f(x) = 0.
The equation given is f(x) = 4x^2 - 7x - 15. To find the x-intercepts, we can set f(x) equal to zero and solve for x.
0 = 4x^2 - 7x - 15
Now, we can factor or use the quadratic formula to solve for x.
Since the quadratic equation does not factor easily, we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
In this equation, a = 4, b = -7, and c = -15.
Using the quadratic formula, we have:
x = (-(-7) ± √((-7)^2 - 4(4)(-15))) / (2(4))
x = (7 ± √(49 - (-240))) / 8
x = (7 ± √(49 + 960)) / 8
x = (7 ± √1009) / 8
So the x-intercepts of the graph of f(x) are (7 + √1009) / 8 and (7 - √1009) / 8.
Part B: To determine whether the vertex of the graph of f(x) is a maximum or minimum, we need to consider the coefficient of the x^2 term in the equation f(x).
In our equation, f(x) = 4x^2 - 7x - 15, the coefficient of x^2 is positive (+4). When the coefficient of x^2 is positive, the parabola opens upward, which means the vertex is a minimum point.
To find the coordinates of the vertex, we can use the formula x = -b / (2a) to find the x-coordinate and substitute it back into the equation to find the y-coordinate.
Using this formula, we have:
x = -(-7) / (2*4)
x = 7/8
Now, substituting this value of x back into the equation f(x), we get:
f(7/8) = 4(7/8)^2 - 7(7/8) - 15
Simplifying this expression, we find:
f(7/8) = 49/8 - 49/8 - 15
f(7/8) = -15
So the coordinates of the vertex are (7/8, -15).
Part C: The steps to graph f(x) are as follows:
1. Plot the x-intercepts: Use the values obtained in Part A, (7 + √1009) / 8 and (7 - √1009) / 8, to plot the points where the graph intersects the x-axis.
2. Plot the vertex: Use the coordinates obtained in Part B, (7/8, -15), to plot the vertex, which is the minimum point of the parabola.
3. Determine the shape of the graph: Since the coefficient of x^2 is positive (+4), the graph opens upward, indicating a U-shaped parabola.
4. Sketch the graph: Connect the plotted points along a smooth curve that passes through the x-intercepts and the vertex. Ensure that the curve maintains the U-shape determined by the coefficient of x^2.
By following these steps, we can accurately graph the function f(x) using the information obtained in Parts A and B.