Question

visual identification
when given the graph of a quadratic, called a _________, the sign of the
discriminant can be determined by looking at the _________ (or - intercepts)
to do this visual inspection you only need to look at the _________ of roots, not the
specific location

(table: if the discriminant is: | number and nature | number of x-intercepts of
the graph of the related
function
positive | two real roots |
zero | one real root (a double root) |
negative | two imaginary roots/ no real
roots |

(table: parabola | number of
roots | sign of
discriminant
(graph 1: parabola crossing x-axis at two points) | 2 |
(graph 2: parabola touching x-axis at one point) | |
(graph 3: parabola not crossing x-axis) | 0 |

Explanation:

Step1: Recall Quadratic Graph Name

A quadratic function's graph is a parabola. So the first blank is "parabola".

Step2: Determine Discriminant Visual Clue

To find the discriminant's sign, we look at the number of x - intercepts (since x - intercepts correspond to real roots). So the second blank is "number" (of x - intercepts).

Step3: Focus on Root Quantity

For visual inspection, we care about the number of roots (not their location). So the third blank is "number".

For the Table:

First Parabola (Two x - intercepts):

• Number of Roots: 2 (two real roots)
• Sign of Discriminant: Positive (since two real roots mean discriminant \(D>0\))

Second Parabola (One x - intercept, vertex on x - axis):

• Number of Roots: 1 (one real root, a double root)
• Sign of Discriminant: Zero (since one real root means discriminant \(D = 0\))

Third Parabola (No x - intercepts):

• Number of Roots: 0 (two imaginary roots, no real roots)
• Sign of Discriminant: Negative (since no real roots mean discriminant \(D<0\))

Answer:

• First blanks: "parabola", "number" (of x - intercepts), "number" (of roots)
• Table:
• First row (two x - intercepts): Number of Roots = 2, Sign of Discriminant = Positive
• Second row (one x - intercept): Number of Roots = 1, Sign of Discriminant = Zero
• Third row (no x - intercepts): Number of Roots = 0, Sign of Discriminant = Negative