The basic quadratic functiony=x2 with the addition of C yields the function y=x2+c The meaning of C is the vertical shift of the function upwards or downwards. If C is positive: the function will rise by the number of steps shown in C. If C is negative: the function will descend by the number of steps shown in C.
Which chart represents the function \( y=x^2-9 \)?
Incorrect
Correct Answer:
4
Question 5
One function
\( y=-2x^2-3 \)
to the corresponding graph:
Incorrect
Correct Answer:
4
Examples with solutions for Parabola of the Form y=x²+c
Exercise #1
One function
y=6x2
to the corresponding graph:
Video Solution
Step-by-Step Solution
The function given is y=6x2. This is a quadratic function, a type of parabola with vertex at the origin (0,0), because there are no additional terms indicating a horizontal or vertical shift.
First, note the coefficient of x2 is 6. A positive coefficient indicates that the parabola opens upwards. The value of 6 means the parabola is relatively narrow, as it is stretched vertically compared to the standard y=x2.
To identify the corresponding graph:
Recognize that a function of the form y=ax2 with a>1 indicates a narrower parabola.
Out of the given graphs, we should look for an upward-opening narrow parabola.
Upon examining each graph, you find that option 2 shows a parabola that is narrower than the standard parabola y=x2 and opens upwards distinctly, matching our function y=6x2.
Therefore, the correct graph for the function y=6x2 is option 2.
Answer
2
Exercise #2
Find the ascending area of the function
f(x)=2x2
Video Solution
Step-by-Step Solution
To determine the intervals where the function f(x)=2x2 is increasing, we will analyze the derivative of the function:
Step 1: Differentiate the function.
The derivative of f(x)=2x2 is f′(x)=4x.
Step 2: Determine where f′(x)>0.
To find the increasing intervals, set 4x>0. Solving this inequality, we obtain x>0.
Therefore, the function f(x)=2x2 is increasing for x>0.
Consequently, the correct answer is the interval where the function is increasing, which is 0<x.
Answer
0 < x
Exercise #3
Find the descending area of the function
f(x)=21x2
Video Solution
Step-by-Step Solution
To solve the problem of finding the descending area of the function f(x)=21x2, we follow these steps:
Step 1: Calculate the derivative of the given function. The function is f(x)=21x2. Differentiating this, we get f′(x)=dxd(21x2)=x.
Step 2: Determine where the derivative is negative. Since f′(x)=x, the derivative is negative when x<0.
Step 3: Conclude the solution. We find that the function f(x) is decreasing for x<0.
Thus, the descending area (domain where the function is decreasing) for the function f(x)=21x2 is x<0.
The correct choice that matches this solution is: x<0.
Answer
x < 0
Exercise #4
Which chart represents the function y=x2−9?
Video Solution
Step-by-Step Solution
To solve the problem of identifying which chart represents the function y=x2−9, let's analyze the function and its graph:
The function y=x2−9 is a parabola that can be described by the general form y=x2+k where k=−9.
It is a standard upward-opening parabola with its vertex located at the point (0,−9). This is because there is no coefficient affecting x, so horizontally it is centered at the origin.
To find the correct graph, we look for one where the bottommost point of the parabola is at (0,−9). This point, known as the vertex, should sit on the y-axis and be the lowest point of the curve due to the upward opening.
After inspecting the charts:
Chart 4 depicts a parabola opening upwards, with its vertex at (0,−9). This aligns perfectly with the form and properties of our function y=x2−9.
Therefore, the chart that represents the function y=x2−9 is Choice 4.
Answer
4
Exercise #5
One function
y=−2x2−3
to the corresponding graph:
Video Solution
Step-by-Step Solution
To solve this problem, we'll match the given function y=−2x2−3 with its corresponding graph based on specific characteristics:
The function y=−2x2−3 is a quadratic equation representing a parabola.
Since the coefficient of x2 is negative, the parabola opens downward.
The y-intercept is -3, which means the parabola crosses the y-axis at −3.
The maximum point (vertex) of the parabola occurs at its axis of symmetry, from which we know it opens downward from that point.
Given these observations, we analyze each graphical option:
Graph 1 represents a parabola opening upward, so it does not match.
Graph 2 might have an appropriate direction but not the correct intercept.
Graph 3 doesn't match key features such as y-intercept and direction.
Graph 4 shows a downward opening parabola with its intercept significantly influenced by negative vertical shift, which matches y=−2x2−3.
Therefore, the function y=−2x2−3 matches with graph option 4.
Answer
4
Question 1
One function
\( y=-6x^2 \)
to the corresponding graph:
Incorrect
Correct Answer:
4
Question 2
One function
\( y=\frac{x^2}{4}+2 \)
to the corresponding graph:
Incorrect
Correct Answer:
1
Question 3
One function
\( y=x^2+9 \)
to the corresponding graph:
Incorrect
Correct Answer:
3
Question 4
Find the corresponding algebraic representation for the function
Incorrect
Correct Answer:
\( y=-x^2+1 \)
Question 5
One function
\( y=-\frac{1}{2}x^2+4 \)
to the corresponding graph:
Incorrect
Correct Answer:
1
Exercise #6
One function
y=−6x2
to the corresponding graph:
Video Solution
Step-by-Step Solution
To solve this problem, we need to match the function y=−6x2 with its graph. This function represents a downward-opening parabola with the vertex at the origin (0,0). The coefficient −6 is negative, confirming it opens downwards, and its large absolute value indicates that the parabola closes towards the axis more sharply than a standard y=−x2 curve.
Let's identify the characteristics of y=−6x2:
- The graph is a parabola, opening downwards.
- The vertex is at the origin, (0,0).
- Symmetric around the y-axis.
- Its steepness is greater than the standard parabola y=−x2 due to the coefficient −6.
By analyzing the given graph options, the graph marked as 4 aligns perfectly with these properties: It is centered on the origin, opens downwards, and has an evident steep slope.
Therefore, the correct graph that matches the function y=−6x2 is option 4.
Answer
4
Exercise #7
One function
y=4x2+2
to the corresponding graph:
Video Solution
Step-by-Step Solution
The function given is y=4x2+2, which is a quadratic function with a vertex at (0,2). The function is in the form y=a(x−h)2+k, where a=41, h=0, and k=2. This tells us that the parabola opens upwards with its vertex at (0,2), and it's wider than the standard parabola y=x2 because 41 is less than 1.
To find the correct graph, look for the one featuring a vertex at (0,2) with an upward opening, and wider spread due to the smaller coefficient. When comparing the graphs, the graph labeled as choice 1 clearly shows these characteristics, indicating the correct match for the function.
Therefore, the solution corresponds to the graph labeled as choice 1.
Answer
1
Exercise #8
One function
y=x2+9
to the corresponding graph:
Video Solution
Step-by-Step Solution
The solution to the problem is choice 3.
Answer
3
Exercise #9
Find the corresponding algebraic representation for the function
Video Solution
Step-by-Step Solution
This problem involves determining the algebraic representation of a parabola that was presented graphically. Our goal is to interpret the graph and express it in terms of its equation for a downward-opening parabola.
To solve the problem, follow these steps:
Step 1: Identify the Parabola's Vertex – According to the diagram, the vertex is positioned at (0,1), implying that at x=0, the maximum value of y is 1. This indicates that the constant term c in the parabola's equation will be 1.
Step 2: Determine the Parabola’s Orientation – The given parabola is described as downward-opening. This means the coefficient in front of x2 must be negative. This leads us to the formula y=−x2+c.
Step 3: Construct the Equation – With the downward orientation and the vertex point established, the equation becomes y=−x2+1 as c=1 from the vertex.
By matching one of the multiple-choice answers with our derived equation, it's clear that choice 2 corresponds to y=−x2+1. Thus
Therefore, the algebraic representation of the function is y=−x2+1.
Answer
y=−x2+1
Exercise #10
One function
y=−21x2+4
to the corresponding graph:
Video Solution
Step-by-Step Solution
To solve for the graph that matches the function y=−21x2+4, let's analyze the function:
The function y=−21x2+4 is a parabola in standard form y=ax2+bx+c, with a=−21, b=0, and c=4.
Because a=−21 is negative, the parabola opens downwards.
The vertex of the parabola y=ax2+bx+c is at x=−2ab. Here, b=0, so x=0.
Substituting x=0 back into the equation gives the vertex's y-coordinate: y=−21(0)2+4=4.
Thus, the vertex is (0,4).
Now, let's match this to the graphs:
We are looking for a graph with a vertex at (0,4) that opens downwards.
Upon reviewing the graphs in the problem, graph number 1 presents a downward opening parabola with a vertex at the point (0,4).
Therefore, the graph that corresponds to y=−21x2+4 is graph 1.
Thus, the solution to the problem is 1.
Answer
1
Question 1
Find the corresponding algebraic representation for the function
Incorrect
Correct Answer:
\( y=x^2-6 \)
Question 2
Find the corresponding algebraic representation for the function
Incorrect
Correct Answer:
\( y=x^2 \)
Question 3
Find the ascending area of the function
\( f(x)=-3x^2+12 \)
Incorrect
Correct Answer:
\( x < 0 \)
Question 4
Find the ascending area of the function
\( f(x)=6x^2-12 \)
Incorrect
Correct Answer:
\( 0 < x \)
Question 5
Find the positive area of the function
\( f(x)=x^2 \)
Incorrect
Correct Answer:
\( x≠0 \)
Exercise #11
Find the corresponding algebraic representation for the function
Video Solution
Step-by-Step Solution
To solve this problem, we will determine the vertical shift given to the parent function y=x2 to form the observed parabola.
Identify that the problem involves a vertical translation of the parabola y=x2.
The function takes the form y=x2+c, where c indicates the vertical shift.
From the graph given, it is seen that the vertex of the parabola is situated at y=−6 when viewed from the intersection with the y-axis.
This downward shift corresponds to the constant c being negative, specifically c=−6.
By this observation, the function becomes y=x2−6.
Therefore, the solution to the problem is y=x2−6, matching choice 2.
Answer
y=x2−6
Exercise #12
Find the corresponding algebraic representation for the function
Video Solution
Step-by-Step Solution
In this problem, we are tasked with identifying the algebraic representation of a function given a graphical depiction. Given the problem's indication that we are dealing with parabolas, particularly those of the form y=x2+c, we need to examine the provided graph for features typical of this family of functions.
The graph structure in the problem suggests a parabolic curve, centered symmetrically, which is indicative of the simplest unmodified parabola, y=x2. The vertex likely lies at the origin, and the parabola opens upwards, a key characteristic of the function y=x2 when the coefficient of x2 is positive and equal to 1.
Upon reviewing the multiple-choice options, the expression that corresponds to this graph is:
Option 1: y=x2
Therefore, the algebraic representation that corresponds to the function is y=x2.
Answer
y=x2
Exercise #13
Find the ascending area of the function
f(x)=−3x2+12
Video Solution
Step-by-Step Solution
Let's solve the problem.
Step 1: Calculate the derivative of the function:
The function is f(x)=−3x2+12.
The derivative f′(x) is calculated using the power rule:
f′(x)=dxd(−3x2+12)=−6x.
Step 2: Find where the derivative is positive:
To find the interval where the function is increasing, solve the inequality f′(x)>0.
This yields:
−6x>0
Divide both sides by −6 (remember to reverse the inequality sign when dividing by a negative):
x<0
Therefore, the function is increasing for x<0.
Thus, the ascending area of the function is x<0.
Answer
x < 0
Exercise #14
Find the ascending area of the function
f(x)=6x2−12
Video Solution
Step-by-Step Solution
To determine the ascending area of the function f(x)=6x2−12, we will follow these steps:
Step 1: Calculate the derivative of the given function.
Step 2: Set the inequality f′(x)>0 to find the interval where the function is increasing.
Step 3: Solve the inequality for x.
Let's begin with Step 1:
The derivative of f(x)=6x2−12 with respect to x is:
f′(x)=dxd(6x2−12)=12x.
Step 2: We need to find where 12x>0. This requires:
x>0.
Step 3: Therefore, the function f(x)=6x2−12 is increasing when x>0.
Thus, the increasing interval of the function is when x>0.
The solution to the problem is 0<x.
Answer
0 < x
Exercise #15
Find the positive area of the function
f(x)=x2
Video Solution
Step-by-Step Solution
To determine where the function f(x)=x2 is positive, we consider the nature of this parabolic graph, which opens upwards.
Step 1: Recognize that the function f(x)=x2 outputs non-negative values for any real number x. The graph of this function is a U-shaped parabola.
Step 2: Analyze the values of the function:
- For x=0, f(0)=02=0.
- For x=0, f(x)=x2>0, because squaring any non-zero real number results in a positive value.
Therefore, the function is positive for all x except at x=0, where it is zero.
Step 3: Based on the comparison given in the choices, and our calculation, the area of interest is positive for x=0.
Thus, the solution to the problem is that the positive area occurs for x=0.