A quadratic function is defined by f(x) = ax^2 + bx + c, where a, b, and c are real consta… — GRE General Test

A quadratic function is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real constants with $a > 0$ . If the vertex of the parabola $y = f(x)$ lies in the third quadrant of the $xy$ -plane, which of the following statements must be true? Select all that apply.

$b > 0$ Cevap
The equation $f(x) = 0$ has two distinct real solutions.Cevap
C
$c$ must be negative.
D
The sum of the two solutions of $f(x) = 0$ is positive.
E
The discriminant $b^2 - 4ac$ is less than zero.

Cevap

The statements '

b > 0

' and 'The equation

f(x) = 0

has two distinct real solutions' must be true.

The vertex of the parabola

y = ax^2 + bx + c

(h, k) = \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)

. Because the vertex is in the third quadrant,

h < 0

and

k < 0

. With

a > 0

, the inequality

-\frac{b}{2a} < 0

directly requires

b > 0

. Furthermore, because

a > 0

(parabola opens upward) and the minimum value

k < 0

lies below the x-axis, the graph must cross the x-axis at two distinct points, establishing that

f(x) = 0

has two distinct real solutions.

Adım Adım Çözüm

Analyze the coordinates of the vertex

(h, k)

relative to the third quadrant.

The third quadrant requires

h < 0

and

k < 0

Points in Quadrant III have negative x-coordinates and negative y-coordinates.

Evaluate the sign of

b

using the x-coordinate formula

h = -\frac{b}{2a}

b > 0

Since

h < 0

and

a > 0

-\frac{b}{2a} < 0 \implies \frac{b}{2a} > 0 \implies b > 0

Determine the number of real solutions using the vertex y-coordinate

k

and orientation

a > 0

The discriminant

b^2 - 4ac > 0

, giving two distinct real solutions.

For an upward-opening parabola (

a > 0

), having a vertex below the x-axis (

k < 0

) guarantees the curve crosses the x-axis twice.

Test counterexamples for

c

and evaluate the root sum

-\frac{b}{a}

c

is not strictly constrained in sign, and the root sum is negative.

The function

f(x) = (x+2)^2 - 1 = x^2 + 4x + 3

has vertex

(-2, -1) \in \text{QIII}

with

c = 3 > 0

. Also, since

a > 0

and

b > 0

, root sum

-\frac{b}{a} < 0

Anahtar Kavram

Parabola Vertex and Quadratic Discriminant Analysis

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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