Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 8
Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=-x^{2}-3$$
Short Answer
Expert verified
The y-intercept is (0, -3), there are no real x-intercepts, and the vertex is (0, -3). The graph is a downward-opening parabola.
Step by step solution
01
Identify the Equation Type
The given equation is a quadratic equation in the form of $$ y = -x^2 - 3 $$This signifies a parabola that opens downward because the coefficient of $$ x^2 $$is negative.
02
Find the Y-Intercept
The y-intercept is found by setting$$ x = 0 $$in the equation. $$ y = -0^2 - 3 $$$$ y = -3 $$So, the y-intercept is $$ (0, -3) $$.
03
Find the X-Intercepts
To find the x-intercepts, set$$ y = 0 $$and solve for$$ x $$.$$ 0 = -x^2 - 3 $$$$ x^2 = -3 $$Since we cannot take the square root of a negative number in the real number system, there are no real x-intercepts.
04
Determine the Vertex
The vertex of a parabola given by$$ y = ax^2 + bx + c $$can be found using the formula$$ x = -\frac{b}{2a} $$. Here,$$ a = -1 $$and$$ b = 0 $$,so$$ x = -\frac{0}{2(-1)} = 0 $$.Plug $$ x = 0 $$into the equation to find$$ y $$.$$ y = -(0)^2 - 3 = -3 $$.Thus, the vertex is$$ (0, -3) $$.
05
Sketch the Graph
Plot the vertex $$ (0, -3) $$and the parabola opening downward. Since there are no x-intercepts, the graph only touches the y-axis at the point$$ (0, -3) $$and extends downward.
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.
quadratic equations
A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). In such an equation, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The quadratic equation forms a parabola when graphed.
Here鈥檚 a breakdown of the elements:
Here鈥檚 a breakdown of the elements:
- \( a \): The coefficient of \( x^2 \), which determines the direction of the parabola's opening.
- \( b \): The coefficient of \( x \), which affects the parabola's symmetry.
- \( c \): The constant term, representing the y-intercept.
parabola
A parabola is the graph of a quadratic equation. It has a symmetrical curve and can open either upwards or downwards. The direction of the parabola depends on the coefficient of the \( x^2 \) term:
A parabola has several important points:
- Upward: If \( a > 0 \), the parabola opens upwards.
- Downward: If \( a < 0 \), the parabola opens downwards.
A parabola has several important points:
- Vertex: The highest or lowest point.
- Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find it, set \( x = 0 \) in the equation and solve for \( y \). For the equation \( y = -x^2 - 3 \):
- When \( x = 0 \), \( y = -0^2 - 3 \)
- Therefore, \( y = -3 \)
- So, the y-intercept is \( (0, -3) \).
x-intercept
An x-intercept is where the graph crosses the x-axis. To find x-intercepts, set \( y = 0 \) and solve for \( x \). For our equation \( y = -x^2 - 3 \):
- Setting \( y = 0 \) gives \( 0 = -x^2 - 3 \).
- Solving for \( x \): \( x^2 = -3 \).
- Since there is no real number whose square is negative, there are no real x-intercepts.
vertex for parabolas
The vertex is an essential feature of a parabola. It represents the maximum or minimum point, depending on whether the parabola opens downward or upward. For the standard equation \( y = ax^2 + bx + c \), the vertex can be found using:
- \( x = -\frac{b}{2a} \)
- So, \( x = -\frac{0}{2(-1)} = 0 \)
- Plugging \( x = 0 \) back into the equation: \( y = -(0)^2 - 3 \)
- This results in \( y = -3 \). So the vertex is at \( (0, -3) \)
graphing
Graphing quadratic equations helps visualize their behavior. To graph the equation \( y = -x^2 - 3 \):
- Start by plotting key points like the y-intercept \( (0, -3) \) and the vertex \( (0, -3) \).
- Next, because \( a = -1 \), the parabola opens downward.
- There are no x-intercepts, so the graph won't cross the x-axis.
