Question

What is the equation of the graph below?

Answer 1

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.

Learn more:

1. Learn more about inverse of the function brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497

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

Answer 2

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).

Learn more

1. What are the coordinates of the vertex of the graph f(x) = x² + 4x − 5? brainly.com/question/1286775
2. What is the y-intercept of the quadratic function  f(x) = (x – 6)(x – 2)? brainly.com/question/1332667
3. Which is the graph of f(x) = (x - 1)(x + 4) brainly.com/question/2334270

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