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HallsofIvy said:The graph is of [tex]y= x^2+ ax+ b[/tex] and we are told that the graph goes through (2, 0). That means that when x= 2, y= 0. So we must have [tex]0= 2^2+ a(2)+ b= 4+ 2a+ b[/tex] or 2a+ b= -4. We are also told that the graph goes through (4, 0). That means that when x= 4, y= 0. So we must have [tex]0= 4^2+ a(4)+ b= 16+ 4a+ b[/tex] or 4a+ b= -16.
Solve the two equations, 2a+ b= -4 and 4a+ b= -16, for a and b.
The y-intercept in a quadratic graph represents the point where the graph intersects with the y-axis. It is also known as the initial value or starting value of the quadratic function. It indicates the value of the dependent variable (y) when the independent variable (x) is equal to 0.
The value of b is the coefficient of the x term in the quadratic function. It determines the slope of the graph and is directly related to the y-intercept. The y-intercept can be calculated by substituting 0 for x in the quadratic function and solving for y.
Yes, the y-intercept can be negative in a quadratic graph. This means that the graph will intersect the y-axis below the origin. The value of the y-intercept will be negative if the quadratic function has a negative constant term (c) or if the value of b is negative.
Changing the value of b will affect the slope of the quadratic graph, which in turn will affect the y-intercept. If b is positive, the graph will have a positive slope and the y-intercept will be above the origin. If b is negative, the graph will have a negative slope and the y-intercept will be below the origin.
The vertex of a quadratic graph is the point where the graph reaches its maximum or minimum value. The y-intercept and the vertex are not always the same point, but they are always on the same vertical line. This means that the x-coordinate of the vertex will be the same as the x-coordinate of the y-intercept.