To determine the signs of the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], and to analyze the equation [tex]\( y = ax^2 + bx + c \)[/tex], let's break down the question step by step.
### Step 1: Understanding the Basic Properties
1. The parabola opens upward.
2. The vertex is in the fourth quadrant.
Properties of the Parabola Equation:
The general form of a parabolic equation is [tex]\( y = ax^2 + bx + c \)[/tex]:
- [tex]\( a \)[/tex]: Determines the direction of the parabola. If [tex]\( a > 0 \)[/tex], the parabola opens upward. If [tex]\( a < 0 \)[/tex], the parabola opens downward.
- [tex]\( b \)[/tex]: Affects the symmetry and tilt of the parabola.
- [tex]\( c \)[/tex]: Represents the y-intercept, i.e., the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
### Step 2: Analyzing the Direction and Location of the Vertex
#### Coefficient [tex]\( a \)[/tex]:
Since the parabola opens upward, [tex]\( a \)[/tex] must be positive.
[tex]\[ a > 0 \][/tex]
#### Vertex in the Fourth Quadrant:
The coordinates of the vertex [tex]\((h, k)\)[/tex] are in the fourth quadrant, meaning [tex]\( h > 0 \)[/tex] and [tex]\( k < 0 \)[/tex].
The x-coordinate of the vertex [tex]\( h \)[/tex] is given by the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Since [tex]\( h > 0 \)[/tex], it follows that:
[tex]\[ -\frac{b}{2a} > 0 \][/tex]
Given [tex]\( a > 0 \)[/tex], to satisfy this inequality, [tex]\( b \)[/tex] must be less than 0.
[tex]\[ b < 0 \][/tex]
#### Coefficient [tex]\( c \)[/tex]:
The fourth quadrant means the vertex is below the x-axis, indicating that:
[tex]\[ k < 0 \][/tex]
The vertex [tex]\( k \)[/tex] of the parabola is calculated with the formula:
[tex]\[ k = c - \frac{b^2}{4a} \][/tex]
Since [tex]\( k < 0 \)[/tex], and knowing that [tex]\( a > 0 \)[/tex] and [tex]\( b < 0 \)[/tex]:
[tex]\[ c - \frac{b^2}{4a} < 0 \][/tex]
To satisfy this innquality, [tex]\( c \)[/tex] has to be smaller than [tex]\( \frac{b^2}{4a} \)[/tex].
### Step 3: Summarizing the Coefficients
- [tex]\( a > 0 \)[/tex]: Positive, since the parabola opens upwards.
- [tex]\( b < 0 \)[/tex]: Negative, based on the direction of the vertex.
- [tex]\( c \)[/tex]: Depends on the equation [tex]\( c < \frac{b^2}{4a}\)[/tex] might be positive or negative.
### Step 4: Possible Graphs:
1. Case 1: [tex]\( c \)[/tex] is positive.
- This means the y-intercept is above the x-axis, while the vertex is below it and the parabola eventually moves upwards past the x-axis.
2. Case 2: [tex]\( c \)[/tex] is negative.
- This means the y-intercept is below the x-axis, and the vertex is also in the fourth quadrant.
### Example Equations and Sketches:
#### Case 1: [tex]\( c \)[/tex] Positive
Example Equation:
[tex]\[ y = 2x^2 - 4x + 1 \][/tex]
#### Case 2: [tex]\( c \)[/tex] Negative
Example Equation:
[tex]\[ y = 2x^2 - 4x - 5 \][/tex]
### Sketches:
For each sketch, focus on:
- The vertex must be in the fourth quadrant.
- The parabola must open upwards.
- In the first case, the y-intercept is above the x-axis.
- In the second case, the y-intercept is below the x-axis.
This analysis gives a comprehensive view of how the coefficients affect the graph and helps understand the nature of the quadratic equation based on given conditions.
