Graph the following quadratic equation by first completing the square: y= −2𝑥+ 6𝑥 + 7.

Question

Graph the following quadratic equation by first completing the square: y= −2𝑥+ 6𝑥 + 7.

Answer 1

-2x^2 + 6x + 7

= -2(x^2-3x) + 7
= -2(x^2 - 3x + 9/4) + 7 + 2(9/4)
= -2(x - 3/2)^2 + 23/2
so the vertex is at (3/2, 23/2) and it opens downward
per the Quadratic Formula, the roots are at
x = (3±√23)/2

Answer 2

To graph the quadratic equation y= −2𝑥+ 6𝑥 + 7 by completing the square, follow these steps:

Step 1: Rearrange the equation so that the x terms are grouped together:
y= -2𝑥+ 6𝑥 + 7
y= (6𝑥 - 2𝑥) + 7
y= 4𝑥 + 7

Step 2: Divide the coefficient of x by 2 and square it. Add this value to both sides of the equation:
y + (4/2)^2 = 4𝑥 + (4/2)^2 + 7
y + 2^2 = 4𝑥 + 2^2 + 7
y + 4 = 4𝑥 + 4 + 7
y + 4 = 4𝑥 + 11

Step 3: Rewrite the equation as a perfect square trinomial on the right side:
y + 4 = 4𝑥 + 11
y + 4 - 11 = 4𝑥
y - 7 = 4𝑥

Step 4: Rewrite the equation in vertex form:
y - 7 = 4𝑥
(y - 7) = 4(x - 0)

Step 5: Identify the vertex of the parabola. The x and y coordinates of the vertex can be determined from the equation in vertex form y = a(x-h)^2 + k.
Comparing the equation y = a(x-h)^2 + k with our equation (y - 7) = 4(x - 0), we can see that h = 0 and k = 7. Therefore, the vertex of the parabola is (0, 7).

Step 6: Plot the vertex on the graph. The vertex represents the minimum or maximum point of the parabola.

Step 7: Find the y-intercept by substituting x=0 into the equation:
y = 4(0) + 7
y = 7

Step 8: Plot the y-intercept on the graph.

Step 9: Find at least two more points on the graph by substituting different x values into the equation. For example, if we let x = 1, we can find the corresponding y value:
y = 4(1) + 7
y = 11

Step 10: Plot the points (1, 11) on the graph.

Step 11: Draw a smooth curve passing through the vertex and the other plotted points. The resulting curve is the graph of the quadratic equation y= −2𝑥+ 6𝑥 + 7.

Please note that the coordinates of the points may vary depending on the accuracy of the graph and scale you are working with.

Answer 3

To graph the given quadratic equation by completing the square, follow these steps:

Step 1: Rearrange the equation
Rearrange the equation in the standard form of a quadratic equation: y = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐. In this case, the equation is:
y = −2𝑥 + 6𝑥 + 7.

Step 2: Group the x terms
Group the x terms together, which gives us:
y = (−2𝑥 + 6𝑥) + 7.

Step 3: Complete the square
To complete the square, divide the coefficient of 𝑥 by 2 and square it. Then, add and subtract this value inside the parentheses. In this case, the coefficient of 𝑥 is 6, so the process will be as follows:
y = (−2𝑥 + 6𝑥 + 9 − 9) + 7.

Step 4: Factor out the common factor and simplify
Rearrange the terms inside the parentheses:
y = (−2𝑥 + 6𝑥 + 9) − 9 + 7.

Simplify inside the parentheses:
y = (4𝑥 + 9) − 2.

Step 5: Identify the vertex
The equation is now in the vertex form: y = 𝑎(𝑥 − ℎ)^2 + 𝑘.
The vertex of the parabola is given by (ℎ, 𝑘). In this case, the vertex is (-9/4, -2).

Step 6: Plot the vertex and find additional points
Using the vertex as a starting point, plot it on the graph. Then, choose some additional x-values, substitute them into the equation, and find the corresponding y-values.
For example, if we choose x = -4, we can substitute it into the equation:
y = (4(-4) + 9) − 2
y = (-16 + 9) − 2
y = -7 - 2
y = -9.

So, when x = -4, y = -9.
Repeat this process for a few more x-values, and plot the corresponding points.

Step 7: Draw the graph
Once you have plotted enough points, connect them with a smooth curve to obtain the graph of the quadratic equation.

Note: The graph may not fit exactly due to the limitations of text-only response. You should try to graph it on a graphing software or draw it on graph paper for more accurate representation.