Draw the graph of the equation 4x²-6x+3 using the -2≤x≤3 use a scale of 2cm to 1unit on the x-axis, 1cm to 4units in the y-axis.
Find the graphfind;
(a) minimum value of y
(b) The roots of the equation 4x²-6x-3=0
(c) The value of x when y=5
(d) The solution set of the equation 4x²-6x-7=0
.
Please help please. I have plot the graph but I dbt know how to do the rest.
did you graph it?
It should be easy to answer some of the questions using the graph.
(a) look at the vertex.
4x^2-6x+3 = (2x-3/2)^2 + 3/4
(b) use the quadratic formula
(c) just solve 4x^2-6x+3=5
(d) again use the QF
Graphic solutions:
Use the graph.
a)
y = 4x²-6x+3
has its mimum value at :
xmin = 3 / 4
ymin = 3 / 4
b)
A vertical translation by 3 - ( - 3 ) = 3 + 3 = 6 units
Subtract 6 to each value of y.
The roots of the equation 4x²-6x-3=0
approx.
x = - 0.4 and x = 1.9
c)
y = 4x²-6x+3 = 5
approx.
x = - 0.3 and x = 1.8
d)
A vertical translation by 3 - ( - 7 ) = 3 + 7 = 10 units
Subtract 10 to each value of y.
The roots of the equation 4x²-6x-7=0
approx.
x = - 0.8 and x = 2.3
To answer the given questions, let's go step by step.
Step 1: Plotting the graph
Use the provided scale to plot the graph of the equation 4x²-6x+3 on the given range of -2≤x≤3. Since the scale is 2cm to 1 unit on the x-axis and 1cm to 4 units on the y-axis, mark intervals of 2 units on the x-axis and 4 units on the y-axis.
The graph should appear as a smooth curve passing through different points. Connect these points with a smooth curve to complete the graph.
Step 2: Finding the minimum value of y (a)
To find the minimum value of y, we need to determine the vertex of the graph. The vertex of a parabola given by the equation ax²+bx+c=0 can be found using the formula x=-b/2a and then substituting this value of x back into the equation to find the corresponding y value.
For our equation, 4x²-6x+3, the coefficient of x² is 4, the coefficient of x is -6, and the constant term is 3. Using the formula, we find x = -(-6)/(2*4) = 3/4.
Now substitute this x value back into the equation to find y:
y = 4(3/4)²-6(3/4)+3 = 9/4 - 18/4 + 12/4 = 3/4.
Therefore, the minimum value of y is 3/4.
Step 3: Finding the roots of the equation 4x²-6x-3=0 (b)
To find the roots of the equation 4x²-6x-3=0, we can use the quadratic formula, x = (-b ± sqrt(b²-4ac))/(2a), where a = 4, b = -6, and c = -3.
Using the formula, we can calculate:
x = (-(-6) ± sqrt((-6)² - 4(4)(-3)))/(2(4))
= (6 ± sqrt(36 + 48))/(8)
= (6 ± sqrt(84))/(8)
= (6 ± 2sqrt(21))/(8)
= (3/4) ± (1/2)sqrt(21).
So, the roots of the equation 4x²-6x-3=0 are (3/4) + (1/2)sqrt(21) and (3/4) - (1/2)sqrt(21).
Step 4: Finding the value of x when y=5 (c)
To find the value of x when y=5, we need to solve the equation 4x²-6x+3=5.
Rearrange the equation to get 4x²-6x-2=0. Now, we can solve this equation using the same quadratic formula as before.
a = 4, b = -6, and c = -2. Using the quadratic formula, we find:
x = (-(-6) ± sqrt((-6)² - 4(4)(-2)))/(2(4))
= (6 ± sqrt(36 + 32))/(8)
= (6 ± sqrt(68))/(8)
= (6 ± 2sqrt(17))/(8)
= (3/4) ± (1/2)sqrt(17).
Hence, the value of x when y=5 is (3/4) + (1/2)sqrt(17) and (3/4) - (1/2)sqrt(17).
Step 5: Finding the solution set of the equation 4x²-6x-7=0 (d)
To find the solution set of the equation 4x²-6x-7=0, follow the same procedure as mentioned above for finding the roots of the quadratic equation.
Here, a=4, b=-6, and c=-7. Using the quadratic formula, we get:
x = (-(-6) ± sqrt((-6)² - 4(4)(-7)))/(2(4))
= (6 ± sqrt(36 + 112))/(8)
= (6 ± sqrt(148))/(8)
= (6 ± 2sqrt(37))/(8)
= (3/4) ± (1/2)sqrt(37).
Therefore, the solution set of the equation 4x²-6x-7=0 is (3/4) + (1/2)sqrt(37) and (3/4) - (1/2)sqrt(37).
By following these steps, you should be able to answer all the questions related to the given graph and equation.