Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 3
Graph each inequality. $$ y > -2 x^{2}-4 x+3 $$
Short Answer
Expert verified
Shade the region above the parabola defined by \( y = -2x^2 - 4x + 3 \).
Step by step solution
01
Identify the inequality and corresponding function
The inequality given is \( y > -2x^2 - 4x + 3 \). This implies that we need to investigate the quadratic function \( y = -2x^2 - 4x + 3 \). Since the inequality symbol is '>', we are interested in the region above this parabola.
02
Determine the vertex of the parabola
The vertex form of a parabola is useful for graphing. The vertex \((h, k)\) of a parabola in standard form \( ax^2 + bx + c \) can be calculated using the formula \( h = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = -4 \). Hence, \( h = -\frac{-4}{2(-2)} = 1 \). To find \( k \), substitute \( x = 1 \) into the equation \( y = -2x^2 - 4x + 3 \). We find \( k = -2(1)^2 - 4(1) + 3 = -3 \). So, the vertex is \( (1, -3) \).
03
Determine the axis of symmetry
The axis of symmetry of the parabola is the vertical line passing through the vertex. For this parabola, the axis of symmetry is \( x = 1 \).
04
Plot the parabola
Using the vertex \((1, -3)\) and the fact that the parabola opens downwards (since \( a = -2 < 0 \)), plot at least a few points on either side of the vertex to sketch the parabola. Choose values of \( x \) such as 0, 2, -1, and find the corresponding \( y \) values to plot the curve.
05
Shade the region above the parabola
Since the inequality is \( y > -2x^2 - 4x + 3 \), we shade the region above the parabola to represent all the points where \( y \) is greater than the value of the quadratic function. Do not include the parabola itself, as this is a strict inequality ('>').
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
Graphing inequalities involves not just plotting a line or curve but also determining the area that satisfies the inequality condition. When you have an inequality such as \( y > -2x^2 - 4x + 3 \), you're essentially looking for all the points where \( y \) is greater than the quadratic expression. To graph this:
- First, plot the corresponding equation \( y = -2x^2 - 4x + 3 \) to find the parabola.
- Use the inequality sign to determine which region to shade. Here, since it's a 'greater than' inequality, you'd shade above the curve.
- Remember not to include the line of the parabola itself since the inequality is strict \( (>) \).
Quadratic Functions
Quadratic functions are essential concepts in algebra represented as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of a quadratic function is a parabola. Unlike linear functions creating straight lines, quadratic functions curve due to the \( x^2 \) term.The parabola can open upwards or downwards:
- If \( a > 0 \), the parabola opens upwards.
- If \( a < 0 \), it opens downwards.
Vertex Form of a Parabola
The vertex form of a parabola provides a clear way to visualize and graph quadratic functions. While the standard form is \( ax^2 + bx + c \), the vertex form is \( a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.To interpret from standard to vertex form, determine the vertex using the formula:
- \( h = -\frac{b}{2a} \)
- \( k \) is found by substituting \( h \) back into the function to get the \( y \)-coordinate of the vertex.
