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Graph the function. g(x)=-(3)/(2)(x-2)^(2) Show Calculator

Graph the function.\newlineg(x)=−32(x−2)2 g(x)=-\frac{3}{2}(x-2)^{2} \newlineShow Calculator

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Full solution

Q. Graph the function.\newlineg(x)=−32(x−2)2 g(x)=-\frac{3}{2}(x-2)^{2} \newlineShow Calculator
  1. Identify Function Type: Identify the type of function. g(x)g(x) is a quadratic function because it's in the form of ax2+bx+cax^2 + bx + c.
  2. Determine Parabola Direction: Determine the direction of the parabola. Since the coefficient of (x−2)2(x - 2)^2 is negative, the parabola opens downwards.
  3. Find Vertex: Find the vertex of the parabola.\newlineThe vertex form of a parabola is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. Here, h=2h = 2 and k=0k = 0, so the vertex is (2,0)(2, 0).
  4. Plot Vertex: Plot the vertex on the graph.\newlineMark the point (2,0)(2, 0) on the coordinate plane.
  5. Identify Axis of Symmetry: Identify the axis of symmetry. The axis of symmetry is x=hx = h, so in this case, x=2x = 2.
  6. Find Additional Points: Find additional points. Choose xx-values around the vertex, like 11 and 33, and plug them into the function to find corresponding yy-values.
  7. Calculate Y-Values: Calculate the y-values for x=1x = 1 and x=3x = 3. For x=1x = 1: g(1)=−(32)(1−2)2=−(32)(−1)2=−(32)(1)=−32g(1) = -\left(\frac{3}{2}\right)(1 - 2)^2 = -\left(\frac{3}{2}\right)(-1)^2 = -\left(\frac{3}{2}\right)(1) = -\frac{3}{2}. For x=3x = 3: g(3)=−(32)(3−2)2=−(32)(1)2=−(32)(1)=−32g(3) = -\left(\frac{3}{2}\right)(3 - 2)^2 = -\left(\frac{3}{2}\right)(1)^2 = -\left(\frac{3}{2}\right)(1) = -\frac{3}{2}.
  8. Plot Additional Points: Plot the points (1,−32)(1, -\frac{3}{2}) and (3,−32)(3, -\frac{3}{2}) on the graph.\newlineThese points will help shape the parabola.
  9. Draw Parabola: Draw the parabola. Connect the points with a smooth curve, making sure it opens downwards and is symmetrical about the line x=2x = 2.

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