Graph, by hand, the function $h ( x )=( x -2)^2+3$.

Question

03-11-2026
Mathematics

Answered

Graph, by hand, the function $h ( x )=( x -2)^2+3$.

Verified Answer :

Related Questions

n a nuclear reaction mass must be lost to produce energy true or false

According to the biological species concept individuals belong to the same species if they can interbreed in nature and produce viable fertile offspring prezygo

Which number is larger 0775 or 095

The creation of the European Union has had all of the following effects except a increased trade between countries b increased immigration to member countries c

a summary about concussions

David opens a bank account with 140 and then deposits 150 each month Louis opens a bank account with 260 and deposits 120 each month How many months will it tak

evaluate 16 23 15 16 1325 156 04 pls answer

What was an immediate cause of the 1789 French Revolution the Enlightenment the Springtime of the Peoples the ideas of theCommunist Manifesto a financial crisis

Write 10 sentences describing your weekend in french

What is the square root of 80 simplified

GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T11:20:32-05:00

Identify the vertex from the form $h(x) = (x - h)^2 + k$ as $(2, 3)$.
Determine that the parabola opens upwards because the lead coefficient is positive.
Calculate key points including the y-intercept $(0, 7)$ and points $(1, 4)$, $(3, 4)$, and $(4, 7)$ to guide the sketch.
Sketch the curve on a coordinate plane, ensuring it is a smooth, symmetric, U-shaped parabola.

Explanation

Analyzing the Function and Identifying the Vertex

The function $h(x) = (x - 2)^2 + 3$ is a quadratic function given in vertex form, $h(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. By comparing the given expression, we identify $h = 2$ and $k = 3$. Therefore, the vertex of this parabola is at the point $(2, 3)$. The coefficient $a = 1$ is positive, which indicates that the parabola opens upwards.

Determining the Axis of Symmetry and Intercepts

The axis of symmetry for a parabola in vertex form is the vertical line passing through the vertex, which is $x = h = 2$. To further define the shape, we calculate the y-intercept by evaluating the function at $x = 0$:
\[h(0) = (0 - 2)^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7.\]
So, the y-intercept is at the point $(0, 7)$. Due to the symmetry of the parabola about the line $x = 2$, the point symmetric to $(0, 7)$ is $(4, 7)$.

Calculating Additional Points for Graphing

To ensure accuracy when sketching the graph, we calculate a few more points around the vertex. For $x = 1$:
\[h(1) = (1 - 2)^2 + 3 = (-1)^2 + 3 = 1 + 3 = 4.\]
By symmetry, at $x = 3$, we also have $h(3) = 4$. Thus, we have the points $(1, 4)$ and $(3, 4)$. We now have a set of points to guide our drawing: $(2, 3)$, $(0, 7)$, $(1, 4)$, $(3, 4)$, and $(4, 7)$.

Sketching the Graph

To graph the function, start by drawing a Cartesian coordinate system with appropriate scales for both the x and y axes. Plot the vertex at $(2, 3)$ and the calculated points $(0, 7)$, $(1, 4)$, $(3, 4)$, and $(4, 7)$. Finally, draw a smooth, continuous, U-shaped curve passing through these points, making sure the curve is symmetric about the vertical line $x = 2$. The resulting graph is a parabola opening upwards with its minimum value at the vertex $(2, 3)$.

Examples

Quadratic functions are used in various real-life scenarios, such as modeling the trajectory of a ball thrown into the air or the path of a projectile. If a person throws a basketball, its height over time can be represented by a quadratic function, similar to the one we just graphed. Understanding how to find the vertex helps in determining the maximum height the ball reaches, while other points help track the ball's position throughout its motion, assisting in sports analytics or physics simulations.