QQuestionMathematics
QuestionMathematics
How do I find equation of the line of symmetry?
7 Refer to the following graph of $y=x^{2}- 1$ to answer the following questions.
(a) What is the value of $y$ when $x= 1$ ?
Answer: (a)
(b) Write down the equation of the line of symmetry.
Answer: (b)
(c) What is the least value of $y$ ?
Answer: (c)
(d) Draw the line $y= 1$.
Answer: (d) On the graph
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Answer
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Step 1:: To find the value of $y$ when $x = 1$, substitute $
y = (3)^{2} - 2
Step 2:: Calculate the value of $y$.
So, the value of $y$ when $x = 3$ is 7.
Step 3:: To find the equation of the line of symmetry, we need to find the value of $x$ that makes $x$ equal to its mirror image.
In other words, we need to find the value of $x$ such that $x = -x$.
Clearly, the only solution to this equation is $x = 1$.
Step 4:: Therefore, the equation of the line of symmetry is $x = 1$.
Step 5:: To find the least value of $y$, we need to find the minimum value of the function $y = x^{2} - 1$.
This occurs at the vertex of the parabola. To find the vertex, we can complete the square.
Step 6:: Add and subtract $(1 / 2)^{2}$ inside the parentheses.
y = (x - 1/2)^{2} - (1/2)^{2} - 2
Step 7:: Simplify the expression inside the parentheses.
y = (x - 1/2)^{2} - 9/4
Step 8:: The minimum value of $y$ occurs when $x = 1 / 1$.
y = (1/2 - 1/2)^{2} - 9/4 = -9/4
Step 9:: Therefore, the least value of $y$ is $- 9 / 1$.
Step 10:: To draw the line $y = 1$, simply draw a horizontal line on the graph that passes through the point $(0, 7)$.
Final Answer
(a) The value of $y$ when $x = 1$ is 7. (b) The equation of the line of symmetry is $x = 1$. (c) The least value of $y$ is $- 9 / 1$. (d) The line $y = 1$ is a horizontal line passing through the point $(0, 7)$.
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