Question

Y=0.5(x+6)^2+8 построить график функций

Answer 1

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Answer 2

To plot the graph of the given function $Y = 0.5(x+6)^2 + 8$, we can follow these steps:

Step 1: Identify the vertex of the parabola. Step 2: Determine the direction of the parabola (opening upwards or downwards). Step 3: Find the y-intercept. Step 4: Find the x-intercepts (if they exist). Step 5: Plot additional points if needed. Step 6: Draw the graph.

Let's start by identifying the vertex. The given function is in the form of $Y = a(x - h)^2 + k$, where (h, k) represents the vertex of the parabola.

Comparing with the given function: $a = 0.5$ $h = -6$ $k = 8$

Step 1: Vertex The vertex is at the point (-6, 8).

Step 2: Direction of the parabola Since the coefficient of $x^2$ is positive (0.5), the parabola opens upwards.

Step 3: y-intercept To find the y-intercept, set $x = 0$ in the equation: $Y = 0.5(0+6)^2 + 8$ $Y = 0.5(6)^2 + 8$ $Y = 0.5(36) + 8$ $Y = 18 + 8$ $Y = 26$

The y-intercept is at the point (0, 26).

Step 4: x-intercepts To find the x-intercepts, set $Y = 0$ in the equation: $0 = 0.5(x+6)^2 + 8$ $-8 = 0.5(x+6)^2$ $-16 = (x+6)^2$

Taking the square root on both sides: $x + 6 = \pm \sqrt{-16}$ $x + 6 = \pm 4i$

Since the solutions are complex numbers ($4i$ and $-4i$), the parabola does not intersect the x-axis, and thus, there are no real x-intercepts.

Step 5: Additional points You can choose some additional values for x and calculate the corresponding Y values to plot more points on the graph.

Let's pick a few x values: When $x = -8$: $Y = 0.5(-8+6)^2 + 8 = 0.5(-2)^2 + 8 = 0.5(4) + 8 = 2 + 8 = 10$

When $x = -4$: $Y = 0.5(-4+6)^2 + 8 = 0.5(2)^2 + 8 = 0.5(4) + 8 = 2 + 8 = 10$

When $x = -2$: $Y = 0.5(-2+6)^2 + 8 = 0.5(4)^2 + 8 = 0.5(16) + 8 = 8 + 8 = 16$

Step 6: Plot the graph Now that we have the vertex and a few additional points, we can plot the graph.

The graph will be a parabola that opens upwards, passing through the vertex (-6, 8), and it won't intersect the x-axis (no real x-intercepts). It will pass through the y-intercept (0, 26).

Note: Since the parabola doesn't have any real x-intercepts, it won't be a full parabola, but rather a portion of it.

Here's the graph:

markdownCopy code
     |   
  30 -|                         * (0, 26)  * 
     |                 *             *
  20 -|              *                 * 
     |             *                     *
  10 -|          *                       *
     |         *                          *
   0 -|       *                            *
     |      *                              *
  -10 -|    *                                *
     |   *                                   *
  -20 -| *                                      *
     |*                                          *
  -30 -|__________________________________________*__
       -10      -5       0       5       10       15

Please note that the graph may not be perfect due to limitations in ASCII art, but it should give you a rough idea of the shape of the parabola. If you have access to graphing software or tools, you can plot the function more accurately.