Q. Which is the graph of \(f(x)=-(x+3)(x+1)\)?

Answer

Given: \( f(x)=-(x+3)(x+1) \)

Zeros: the roots are \(x=-3\) and \(x=-1\), i.e. the points \((-3,0)\) and \((-1,0)\).
Expansion / leading coefficient:
\[
f(x)=-(x+3)(x+1)=-(x^2+4x+3)=-x^2-4x-3.
\]
The leading coefficient is \(-1\), so the parabola opens downward.
Axis of symmetry:
\[
x=\frac{-3+(-1)}{2}=-2.
\]
Vertex:
\[
f(-2)= -\bigl((-2+3)(-2+1)\bigr)=-(1\cdot(-1))=-(-1)=1,
\]
so the vertex is \((-2,1)\).
Y-intercept:
\[
f(0)=-(0+3)(0+1)=-(3\cdot1)=-3,
\]
so the y-intercept is \((0,-3)\).

Final:
\[
\boxed{\text{Downward-opening parabola with roots }(-3,0)\text{ and }(-1,0),\ \text{vertex }(-2,1),\ \text{y-intercept }(0,-3).}
\]

Detailed Explanation

Problem: Graph the quadratic function \( f(x) = -(x + 3)(x + 1) \)

Step 1 – Direction of opening

The factored form is \( f(x)=a(x-p)(x-q) \) with \( a=-1 \). Since the leading coefficient \( a \) is negative, the parabola opens downward. (Expanded form: \( f(x)=-(x^{2}+4x+3)=-x^{2}-4x-3 \).)

Step 2 – x-intercepts (roots)

Set \( f(x)=0 \) and solve each factor:

\[
x+3=0 \Rightarrow x=-3
\]

\[
x+1=0 \Rightarrow x=-1
\]

Thus the graph crosses the x-axis at the points \( (-3,0) \) and \( (-1,0) \).

Step 3 – y-intercept

Evaluate at \( x=0 \):

\[
f(0)=-(0+3)(0+1)=-(3)(1)=-3
\]

So the y-intercept is \( (0,-3) \).

Step 4 – Vertex

The x-coordinate of the vertex is the midpoint of the roots:

\[
x=\frac{-3+(-1)}{2}=\frac{-4}{2}=-2
\]

Evaluate \( f \) at \( x=-2 \):

\[
f(-2)= -(-2+3)(-2+1)= -(1)(-1)=1
\]

The vertex is \( (-2,1) \). The axis of symmetry is \( x=-2 \) and, because the parabola opens downward, the vertex is a maximum with value \( 1 \).

Conclusion

The graph is a downward-opening parabola with:

vertex at \( (-2,1) \),
x-intercepts at \( (-3,0) \) and \( (-1,0) \),
y-intercept at \( (0,-3) \),
axis of symmetry \( x=-2 \).

See full solution

Graph

Get step-by-step help for quadratic graphs.

AI for homework

FAQs

What are the x-intercepts (zeros) of (f(x)=-(x+3)(x+1))?

Solve ((x+3)(x+1)=0). Zeros: (x=-3) and (x=-1). Points: ((-3,0)) and ((-1,0)).

What is the y-intercept of (f)?

Evaluate (f(0)=-(0+3)(0+1)=-3). Y-intercept: ((0,-3)).

Does the parabola open up or down; what is the leading coefficient?

Expanding gives (f(x)=-x^2-4x-3). Leading coefficient (-1), so the parabola opens downward.

Where is the vertex and is it a max or min?

Vertex x is midpoint of roots: (x=-2). Compute (f(-2)=1). Vertex ((-2,1)); it is a maximum of (1).

What is the axis of symmetry?

The axis is the vertical line (x=-2).

What is the vertex form and what transformations from (y=-x^2)?

Vertex form: (f(x)=-(x+2)^2+1). Transform: reflect over x-axis, shift left 2, then up 1.

What are the domain and range of (f)?

Domain: all real numbers. Range: (yle 1) (since maximum value is (1)).

On which intervals is (f) increasing or decreasing?

Increasing on ((-infty,-2)). Decreasing on ((-2,infty)).

How to sketch the graph quickly?

Plot vertex ((-2,1)), zeros ((-3,0)), ((-1,0)), and y-intercept ((0,-3)). Draw a downward parabola symmetric about (x=-2).

How does the parabola's width compare to (y=-x^2)?

Same width (same steepness), because the leading coefficient magnitude is (1).

Discover new AI picks today.
Try one today now

173,935+ happy customers
Math, Calculus, Geometry, etc.

Linear Algebra Solver Math AI Solver Calculus AI