Solving Quadratic functions?

Question

Graphing with standard form. Graph the Function. Label the vertex and the axis of symmetry

y=2x^2-12x+4

Answer 1

Graph of the parabola in standard form :

The function is f(x) = y = 2x² - 12x + 4.

The standard form of parabola equation is f(x) = ax² + bx + c.

Step 1 :

Find the axis of symmetry :

Formula for the equation of the axis of symmetry:  x = - b/2a.

Substitute the values of b = - 12 and a =  2 in the formula, x = - b/2a.

x = - (- 12)/2( 2)

x = 12/4 = 3.

The equation for the axis of symmetry is x = 3.

Step 2 :

Find the vertex :

To find the vertex, use the value of equation for the axis of symmetry as the  x - coordinate of the vertex.

To find the y - coordinate, substitute the value of x = 3 in the original equation, y = 2x² - 12x + 4.

y =  2( 3)² - 12(3) + 4

y = 18 - 36 + 4

y = -14.

The vertex is (3, -14).

Determine whether the function has maximum or minimum value :

The value of a = 2 > 0 (positive), so the graph of function opens upward and has a minimum value.

The minimum value (y - coordinate of the vertex) is -14.

Step 3 :

Find the y -  intercept :

To find the y - intercept, Substitute the value x = 0 in the original function, f(x) = y = 2x² - 12x + 4.

y = 2(0)² - 12(0) + 4

y = 4.

The y - intercept is 4.

Step 4 :

• Lets find the another point.Choose an x - value of 1 and substitute.The new point is at (1, -6).
• The point paired with it on other side of the axis of symmetry is (5, - 6) and has the same y - value.
• Since, The axis of symmetry devides the parabola into two equal parts.So, if there is a point (1, - 6) on one side, there is a corresponding point on other side that is the same distance from the axis of symmetry and has the same y - value.
• The distance between the points (1, - 6) and (5, - 6) = 4 = The distance between ( 5, - 6) and the point paired with it on other side of the axis of symmetry and has the same y - value.
• The distance between (5, - 6) and the point paired with it on other side of the axis of symmetry. = 5 - 3 = 2.
• Therefore, The point paired with it on other side of the axis of symmetry is (5, - 6).
• The y - intercept is (0, 4), so the point paired with it on other side of the axis of symmetry is (6, 4) and has the same y - value.
• Therefore, The point paired with it on other side of the axis of symmetry is ( 6, 4).
• Connect these points and create a smooth curve.

Graph :