Question

Which statements about the behavior of f(x) = x2 + 4x − 3 are true? Select all that apply.

A. f has an axis of symmetry at x = –2.
B. The range of f is the set of all real numbers.
C. The maximum value of f is –7 when x = –2.
D. As x approaches infinity, f(x) approaches infinity.
E. As x approaches negative infinity, f(x) approaches negative infinity.

Answer 1

A, D

Explanation:

Given function:
`f(x)=x^2+4x-3`

Axis of symmetry

`\textsf{Axis of Symmetry formula : }x=-(b)/(2a)`

for a quadratic equation in standard form
`y=ax^2+bx+c`

`\implies \textsf{Axis of symmetry}: x=-(4)/(2)=-2`

Maximum/Minimum point (vertex)

The max/min point is the turning point of the parabola.

The x-value of the turning point is the axis of symmetry.

`\implies \textsf{Turning point}:f(-2)=(-2)^2+4(-2)-3=-7`

Turning point (vertex) = (-2, -7)

As the leading coefficient is positive, the parabola opens upwards. Therefore, the turning point (-2, -7) will be a minimum.

Domain & Range

Domain: input values → All real numbers

Range: output values →
`x\geq -7` [as (-2, -7) is the minimum point]

End behavior

As the leading degree is positive and the leading coefficient is positive:

`f(x) \rightarrow + \infty, \textsf{ as } x \rightarrow - \infty`

`f(x) \rightarrow + \infty, \textsf{ as } x \rightarrow + \infty`