Question

1 which quadratic function graphed below has a vertex at (3, -2)?
(a) (b)
(c) (d)

1. what is the domain and range of the quadratic function graphed?

(a) domain: {x | x∈r} range: {y | y≤-4; y∈r}
(b) domain: {x | x∈r} range: {y | y≥-4; y∈r}
(c) domain: {x | x≤-3 ; x∈r} range: {y | y∈r}
(d) domain: {x | x≥-3 ; x∈r} range: {y | y∈r}

1. what is the smallest value of x that satisfies this equation x(x+4) = -3

a -1
b 0
c 1
d -3

1. which of the following is the graph of f(x) = (x - 1)(x - 5)?

a) b) c) d)

Explanation:

Question 1

Step1: Recall Vertex Definition

The vertex of a parabola (quadratic function graph) is its minimum (for upward - opening) or maximum (for downward - opening) point. We need to find which graph has its vertex at \((3,-2)\), i.e., the point where \(x = 3\) and \(y=-2\).

Step2: Analyze Each Graph

• For graph A: Check the coordinates of the vertex. By looking at the grid, the vertex seems to be at \((3,-2)\) (since the lowest point of the parabola is at \(x = 3\) and \(y=-2\)).
• For graph B: The vertex is at a different \(x\) - coordinate (not 3) when we check the grid.
• For graph C: The vertex has a negative \(x\) - coordinate, not 3.
• For graph D: The vertex has a negative \(x\) - coordinate, not 3.

Step1: Recall Domain and Range of Quadratic Functions

The domain of a quadratic function \(y = ax^{2}+bx + c\) (where \(a
eq0\)) is all real numbers, i.e., \(\{x|x\in R\}\) because we can plug in any real number for \(x\). The range depends on the direction the parabola opens. If the parabola opens upward (\(a>0\)), the range is \(\{y|y\geq k\}\) where \(k\) is the \(y\) - coordinate of the vertex. If it opens downward (\(a < 0\)), the range is \(\{y|y\leq k\}\).

Step2: Analyze the Given Graph

The given parabola opens upward (since it has a minimum point). The vertex of the parabola (the minimum point) has a \(y\) - coordinate of \(-4\). So the domain is all real numbers (\(\{x|x\in R\}\)) and the range is \(\{y|y\geq - 4;y\in R\}\), which corresponds to option B.

Step1: Rewrite the Equation

Start with the equation \(x(x + 4)=-3\). Expand the left - hand side: \(x^{2}+4x=-3\). Then, rewrite it in standard quadratic form \(ax^{2}+bx + c = 0\) by adding 3 to both sides: \(x^{2}+4x + 3=0\).

Step2: Factor the Quadratic Equation

Factor \(x^{2}+4x + 3\). We need two numbers that multiply to 3 and add up to 4. The numbers are 1 and 3. So, \(x^{2}+4x + 3=(x + 1)(x+3)=0\).

Step3: Solve for \(x\)

Set each factor equal to zero: \(x + 1=0\) gives \(x=-1\) and \(x + 3=0\) gives \(x=-3\).

Step4: Find the Smallest Value

Among \(-3\) and \(-1\), the smallest value is \(-3\).

Answer:

A

Question 2