The graph of a quadratic function can have 0, 1 or 2 x-intercepts. How can you predict the number of x-intercepts without drawing the graph or (completely) solving the related equation?

Suppose that the graph of f(x) = ax^2+bx+c has x-intercepts (m,0) and (n,0). What are the x-intercepts of g(x) = –ax^2–bx–c?

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1. - a x ^ 2 - b x - c = - ( a x ^ 2 + b x + c )

Quadratic function can be concave up or concave down.

If a x ^ 2 + b x + c concave up or concave down graph of - a x ^ 2 - b x - c will be concave down or concave up.

The graph is rotated 180 ° around the x - axis, but x-intercepts stay same.

So x-intercepts of - a x ^ 2 - b x - c = x-intercepts of a x ^ 2 + b x + c

( m , 0 ) , ( n , 0 )

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2. How can you predict the number of x-intercepts without drawing the graph.

Discriminant = b ^ 2 - 4 a c

If the discriminant is positive, then there are two distinct roots ( x - intercepts ).

If the discriminant is zero, then there is exactly one real root x - intercept ).

If the discriminant is negative, then there are no real roots( no one x - intercepts ).

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3. x intercepts -3,1 points on the graph (
2,2.5)

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