To put the equation y = -4x^2 + 48x - 149 into graphing form, you will need to complete the square. Here is a step-by-step process:
Begin by factoring out the coefficient of the x^2 term. In this case, the coefficient is -4, so you will need to factor out a -4:
y = (-4)(x^2) + 48x - 149
Next, you will need to add and subtract a value in order to complete the square. The value you add and subtract should be equal to the square of half of the coefficient of the x term. In this case, the coefficient of the x term is 48, so you will need to add and subtract (48/2)^2 = 144:
y = (-4)(x^2 + (48/2)^2 - (48/2)^2) + 48x - 149
Simplify the expression inside the parentheses:
y = (-4)(x^2 + 144 - 144) + 48x - 149
Simplify the expression inside the parentheses:
y = (-4)(x^2) + 48x - 149
Rearrange the terms so that the x^2 term is on the left side and the constant term is on the right side:
x^2 - 12x - 37 = 0
Factor the quadratic equation:
(x - 7)(x + 5) = 0
The solutions to the equation are x = 7 and x = -5. These are the x-coordinates of the points where the graph of the equation intersects the x-axis.
To plot these points on the graph, you will need to substitute each value of x into the original equation and solve for y. For example, if x = 7, then y = (-4)(7^2) + 48(7) - 149 = 25. Similarly, if x = -5, then y = (-4)(-5^2) + 48(-5) - 149 = -149.
Plot the points (7, 25) and (-5, -149) on the graph. The graph of the equation y = -4x^2 + 48x - 149 will be the parabola that passes through these two points.