Answer:
The maximum value occurs at point (-2, 8) where x = -2 represents the axis of symmetry.
Step-by-step explanation:
<h3>General Concepts: </h3>
Quadratic functions.
Vertex form.
Standard form.
Axis of symmetry.
Maximum value.
Minimum value.
BPEMDAS Order of Operations:
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<h2>Vertex Form:</h2>
The vertex form of a quadratic function is f(x) = a(x - h)² + k, where <em>a</em> ≠ 0 and (<em>h, k </em>) represents the coordinates of the vertex.
a: Vertical stretch or compression factor.
- a > 0 ⇒ The graph opens <u>upward</u>, and the y-coordinate of the vertex represents the minimum value.
- a < 0 ⇒ The graph opens <u>downward</u>, and the y-coordinate of the vertex represents the maximum value.
h: Horizontal translation.
- h > 0 ⇒ The graph shifts <em>"h"</em> units to the right.
- |h| < 0 ⇒ The graph shifts <em>"h" </em>units to the left.
k: Vertical translation.
- k > 0 ⇒ The graph shifts <em>"k"</em> units upward.
- |k| < 0 ⇒ The graph shifts <em>"k"</em> units downward.
<h3>Axis of symmetry:</h3>
The axis of symmetry is an imaginary vertical line that goes through the vertex of a parabola and divides the graph into two symmetrical halves. The axis of symmetry is also the <u>x-coordinate</u> of the vertex, (h, k). Hence, the axis of symmetry is: <em>x </em>= <em>h</em>.
<h3>Find the axis of symmetry:</h3>
Step 1: Transform the given quadratic function into its general form, h(x) = ax² + bx + c.
Given: h(x) = - (x + 2)² + 8
Step 2: Solve for the axis of symmetry.
Substitute the derived values for <em>a</em> = -1 and <em>b</em> = -4 into the following formula:
Step 3: Find the <u>maximum value</u>.
Substitute the derived value for the axis of symmetry, x = -2, into h(x) = -x² - 4x + 4.
h(x) = -x² - 4x + 4 ⇒ General form.
Hence, the maximum value (vertex) occurs at point (-2, 8).
<h2>Graph the parabola: </h2><h3>
Find other points to plot:</h3>
In order to find other points to plot on the graph, we can substitute different x-values into the vertex form. A good starting point is to solve for the y-intercept, which is the point on the graph where it intersects the y-axis.
<h3>Solve for the y-intercept:</h3>
Solve for the y-intercept by setting x = 0:
Vertex form: h(x) = - (x + 2)² + 8
⇒ h(0) = -(0 + 2)² + 8 → [Let <em>x</em> = 0 in h(x) = - (x + 2)² + 8].
⇒ h(0) = -(2)² + 8 → [PEMDAS: Parenthesis].
⇒ h(0) = -4 + 8 → [PEMDAS rule: Exponent].
⇒ h(0) = 4 → [PEMDAS: Addition].
Hence, the y-intercept is (0, 4).
<h3>Graph the axis of symmetry:</h3>
We can graph the <em>axis of symmetry</em> by drawing a <u>vertical line</u> from x = -2, and use it as a reference point in finding other points to plot on the graph.
Since the y-intercept, (0, 4) is 2 units to the right of the <em>axis of symmetry</em>, then it means that going 2 units to the left will give us (-4, 4).
We have the following points to plot on the graph:
- Vertex (maximum value): (-2, 8)
- Axis of symmetry: x = -2.
- Y-intercept: (0, 4).
- Other point: (-4, 4).
<h2>Final Answer: </h2>
The maximum value occurs at point (-2, 8), where <em>x</em> = -2 represents the axis of symmetry.
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Learn more about quadratic functions here:
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