Question

19 iven the following graph of the quadratic function $f(x)=\frac{1}{3}x^2 - 3x + 9$, determine whether the discriminant of the equation is positive, negative, or zero.

Explanation:

Step1: Recall discriminant - graph relation

For a quadratic function \( y = ax^2 + bx + c \), the discriminant is \( D = b^2 - 4ac \). The graph's intersection with the \( x \)-axis depends on \( D \):

• If \( D>0 \): Two distinct \( x \)-intercepts (crosses \( x \)-axis twice).
• If \( D = 0 \): One \( x \)-intercept (touches \( x \)-axis once).
• If \( D<0 \): No \( x \)-intercepts (does not cross \( x \)-axis).

Step2: Analyze the given graph

The graph of \( f(x)=\frac{1}{3}x^2 - 3x + 9 \) is a parabola opening upwards (since the coefficient of \( x^2 \) is positive) and it does not intersect the \( x \)-axis (as seen from the graph, the parabola is above the \( x \)-axis).

Step3: Determine the discriminant

Since the graph has no \( x \)-intercepts, by the relation between discriminant and \( x \)-intercepts, the discriminant \( D=b^2 - 4ac \) for \( ax^2+bx + c = 0 \) (here \( a=\frac{1}{3} \), \( b=-3 \), \( c = 9 \)) must be negative.

Answer:

The discriminant is negative.