Graph the equation y, equals, minus, x, squared, plus, 12, x, minus, 32y=−x 2 +12x−32 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

Sure, here are the steps to graph the equation y = -x^2 + 12x - 32:

Step 1: Find the roots of the equation The roots of the equation are the x-values where y = 0. To find the roots, set the equation to 0 and solve for x:

0 = -x^2 + 12x - 32

This is a quadratic equation, and you can solve it using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a. In this case, a = -1, b = 12, and c = -32.

Step 2: Find the vertex of the parabola The vertex of a parabola y = ax^2 + bx + c is given by the point (h, k), where h = -b/2a and k = f(h). In this case, h = -12/2*(-1) = 6. To find k, substitute h = 6 into the equation to get k = -(6)^2 + 12*6 - 32 = -4.

Step 3: Choose two more points to plot Now that you have the roots and the vertex, choose two more points to plot. You can choose any x-values you like. For example, you might choose x = 0 and x = 10. Substitute these values into the equation to find the corresponding y-values.

Step 4: Plot the points on the graph Now that you have five points (the roots, the vertex, and the two points you chose), plot these points on the graph.

Step 5: Draw the parabola Finally, draw the parabola that passes through the five points. The parabola should be a downward-opening curve because the coefficient of x^2 is negative.

Remember, the exact values of the roots and the vertex will depend on the solutions to the equations in steps 1 and 2.

Sure, let's graph the equation y = -x^2 + 12x - 32 step by step.

Step 1: Identify the roots of the equation The roots of the equation are the x-values where y = 0. We can find these by setting the equation equal to zero and solving for x:

0 = -x^2 + 12x - 32

This is a quadratic equation in the form ax^2 + bx + c = 0. We can solve it using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a):

x = [-12 ± sqrt((12)^2 - 4*(-1)(-32))] / (2(-1)) x = [-12 ± sqrt(144 - 128)] / -2 x = [-12 ± sqrt(16)] / -2 x = [-12 ± 4] / -2

So the roots of the equation are x = 8 and x = 4. These are two points we can plot on the graph.

Step 2: Identify the vertex of the equation The vertex of a parabola y = ax^2 + bx + c is given by the point (h, k), where h = -b/(2a) and k = f(h).

h = -b/(2a) = -12/(2*-1) = 6 k = f(h) = -6^2 + 12*6 - 32 = -36 + 72 - 32 = 4

So the vertex of the equation is (6, 4). This is another point we can plot on the graph.

Step 3: Choose two more points to plot We have already plotted the roots and the vertex, so we just need to choose two more points. It's usually easiest to choose points close to the vertex. Let's choose x = 5 and x = 7.

When x = 5, y = -5^2 + 125 - 32 = -25 + 60 - 32 = 3 When x = 7, y = -7^2 + 127 - 32 = -49 + 84 - 32 = 3

So the points (5, 3) and (7, 3) are also on the graph.

Step 4: Plot the points and draw the graph Now we can plot the points (4, 0), (8, 0), (6, 4), (5, 3), and (7, 3) on the graph. Since we know the graph is a parabola that opens downwards (because the coefficient of x^2 is negative), we can draw a smooth curve through the points to complete the graph.