Question

1. which of the following best represents the intercepts of the graph shown below? a. x-intercept: (0,-6) y-intercepts: (7,0) and (1,0) b. x-intercepts: (0,-7) and (0,-1) y-intercept: (7,0) c. x-intercept: (-6,0) y-intercepts: (0,7) and (0,1) d. x-intercepts: (-7,0) and (-1,0) y-intercept: (0,7) 28. factor the following expression: $x^2 - 7x + 12$. f. $(x + 3)(x + 4)$ g. $(x - 3)(x - 4)$ h. $(x + 1)(x + 12)$ j. $(x - 5)(x - 2)$ 29. which graph represents the function $y = \frac{-1}{2}(x + 2)^2 + 4$? (a) (b) (c) (d)

Explanation:

Question 27

Step1: Recall Intercept Definitions

x - intercepts are points where \( y = 0 \) (on the x - axis), and y - intercepts are points where \( x = 0 \) (on the y - axis).

Step2: Analyze the Graph

Looking at the graph, the parabola crosses the x - axis at \( x=-7 \) and \( x = - 1 \), so the x - intercepts are \( (-7,0) \) and \( (-1,0) \). It crosses the y - axis at \( y = 7 \), so the y - intercept is \( (0,7) \).

Step3: Evaluate Options

• Option A: x - intercept should have \( y = 0 \), but \( (0,-6) \) has \( x = 0 \), wrong.
• Option B: x - intercepts should have \( y = 0 \), \( (0,-7) \) and \( (0,-1) \) have \( x = 0 \), wrong.
• Option C: x - intercept \( (-6,0) \) does not match the graph, wrong.
• Option D: Matches the x - intercepts \( (-7,0) \) and \( (-1,0) \) and y - intercept \( (0,7) \).

Step1: Recall Factoring Quadratics

For a quadratic \( x^{2}+bx + c \), we need two numbers that multiply to \( c \) and add to \( b \). For \( x^{2}-7x + 12 \), we need two numbers that multiply to 12 and add to - 7.

Step2: Find the Numbers

The numbers - 3 and - 4 multiply to \( (-3)\times(-4)=12 \) and add to \( - 3+( - 4)=-7 \).

Step3: Factor the Quadratic

So \( x^{2}-7x + 12=(x - 3)(x - 4) \).

Step1: Recall Vertex Form of a Parabola

The vertex form of a parabola is \( y=a(x - h)^{2}+k \), where \((h,k)\) is the vertex and \( a \) determines the direction and width. For \( y=\frac{-1}{2}(x + 2)^{2}+4 \), the vertex is \((-2,4)\), and \( a=-\frac{1}{2}<0 \), so the parabola opens downward.

Step2: Analyze the Vertex and Direction

• The vertex is at \((-2,4)\).
• Option A: The vertex seems to be at \((-2,4)\) (by looking at the graph's peak), and it opens downward.
• Option B: Vertex is not at \((-2,4)\), wrong.
• Option C and D: Open upward (\( a>0 \) - like shape), but our \( a=-\frac{1}{2}<0 \), so they are wrong.

Answer:

D. x - intercepts: \((-7,0)\) and \((-1,0)\)
y - intercept: \((0,7)\)

Question 28