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What is the equation of the graph below?


A graph shows a parabola that opens up and crosses the x axis near one and a half and four and a half.


Answers:


y = − (x − 3)2 − 2


y = − (x + 2)2 − 3


y = (x − 3)2 − 2


y = (x + 2)2 − 3

Verified Answer :

BladeRunner212

C). y = (x − 3)² − 2

Further explanation

We will solve the problems associated with the quadratic functions.

Given:

A graph shows a parabola that opens up and crosses the x-axis near one and a half and four and a half.

Question:

What is the equation of the graph below?

A) y = − (x − 3)² − 2

B) y = − (x + 2)² − 3

C) y = (x − 3)² − 2

D) y = (x + 2)² − 3

The Process:

If we know the vertex point of the parabola, V(h, k), then we use the formula: \(\boxed{\boxed{ \ y = a(x - h)^2 + k \ }}\)

When \(\boxed{ \ a > 0 \ }\), the parabola opens upward. Therefore we have C or D as a candidate for the answer.

The axis of symmetry is the line that passes through the vertex of a parabola with equation: \(\boxed{ \ x = h = - \frac{b}{2a} \ }\)

If (x₁, 0) and (x₂,0) are the x-intercepts of the graph, then the axis of symmetry can also be determined by \(\boxed{ \ x = h = \frac{x_1 + x_2}{2} \ }\)

From the problem, the graph crosses the x-axis near one and a half and four and a half. So we can find the axis of the symmetry.

\(\boxed{ \ x = h = \frac{1.5 + 4.5}{2} \ }\)

\(\boxed{ \ h = 3 \ }\)

Thus, we have found the equation of the graph, which is \(\boxed{\boxed{ \ C) \ y = (x - 3)^2 - 2 \ }}\).

From the equation of the graph, we know that the vertex point of the parabola is (h, k) = (3, -2), resulting in a vertex that is a minimum.

See the attachment.

- - - - - - - - - -

Notes:

Let us change to standard equation form.

y = (x − 3)² − 2

y = x² - 6x + 9 - 2

y = x² - 6x + 7

A quadratic function is described by the standard equation \(\boxed{\boxed{ \ f(x) = ax^2 + bx + c \ }}\), and the value c is the y-intercept of the graph.

See the attachment once again, the y-intercept is (0, 7).

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Keywords: a graph, shows a parabola, that opens up, crosses the x-axis, near one and a half, four, the quadratic functions, the equation of the graph, vertex, the axis of symmetry, opens upward, the x-intercept, the y-intercept, minimum

View image BladeRunner212

The equation of the graph is \(\boxed{y = {{\left( {x - 3} \right)}^2} - 2}.\)Option (c) is correct.


Further explanation:

The general equation of the parabola can be expressed as follows,

\(\boxed{y = a{{\left( {x - h} \right)}^2} + k}\)


Here, h and k the vertices.


Given:

The equations are as follows,

(a). \(y =  - {\left( {x - 3} \right)^2} - 2\)

(b). \(y =  - {\left( {x + 2} \right)^2} - 3\)

(c). \(y = {\left( {x - 3} \right)^2} - 2\)

(d). \(y = {\left( {x + 2} \right)^2} - 3\)


Explanation:

If the value of a is greater than zero than the parabola is an upward.

If the value of a is less than zero than the parabola is downward.


The value of a in equation (a) and equation (b) is less than zero. Therefore, option (a) and option (b) is not correct.


The x-intercepts are 1.5 and 4.5.

The value of h can be obtained as follows,

\(\begin{aligned}h&= \frac{{1.5 + 4.5}}{2}\\&= \frac{6}{2} \\ &= 3\\\end{aligned}\)


Hence, the equation is \(y = {\left( {x - 3} \right)^2} - 2.\)


The equation of the graph is \(\boxed{y = {{\left( {x - 3} \right)}^2} - 2}\). Option (c) is correct.


Option (a) is not correct.

Option (b) is not correct.

Option (c) is correct.

Option (d) is not correct.


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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Polynomials


Keywords: parabola, graph, shows, opens up, crosses x-axis, one, one and a half, and four and a half, quadratic equation, equation factorization. Factorized form, polynomial, quadratic formula, zeroes, Fundamental Theorem of algebra, polynomial

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