Answer:
Part A: To completely factor f(x) = 2x^2-
5x + 3, we need to break down the
quadratic expression into its factors. The factored form of the quadratic equation is given by: f(x) = (2x-1)(x-3)
Part B: To find the x-intercepts of the graph of f(x), we set f(x) = 0 and solve for
X:
(2x-1)(x-3)=0
Setting each factor equal to zero:
2x-1=0
x-3=0
Solving these equations, we find: 2x=1--> x=1/2
X=3
Therefore, the x-intercepts of the graph of f(x) are x = 1/2 and x = 3.
Part C: The end behavior of the graph of f(x) can be determined by looking at the leading term, which is 2x^2. As the coefficient of the leading term is positive, it indicates that the graph opens upward. This means that as x approaches positive
or negative infinity, the function f(x) also increases without bound.
Part D: To graph f(x), we can utilize the answers obtained in Part B and Part C.
1. Plot the x-intercepts: Mark the points (1/2, 0) and (3,0) on the x-axis. 2. Consider the end behavior: As x approaches positive or negative infinity, the graph increases without bound in an upward direction.
3. Determine the vertex: The vertex of a quadratic function can be found using the formula x = -b/(2a), where a and b are coefficients of the quadratic expression. In this case, a = 2 and b = -5. Calculating the vertex, we find x=-
(-5)/(2*2)=5/4. Plugging this x-value back into the equation, we can find the corresponding y-value: f(5/4) = 2(5/4)^2-5(5/4)+3=1/8. Thus, the vertex is approximately (5/4, 1/8).
. Sketch the graph: Using the x- intercepts, the end behavior, and the vertex, we can draw the graph of f(x) accordingly. The graph should be a U- shaped curve opening upward, passing through the x-intercepts, and with the vertex as the lowest point.
Explanation: