Answer:
The maximum value occurs at point (-2, 8) where x = -2 represents the axis of symmetry.
Step-by-step explanation:
General Concepts:
Quadratic functions.
Vertex form.
Standard form.
Axis of symmetry.
Maximum value.
Minimum value.
BPEMDAS Order of Operations:
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Vertex Form:
The vertex form of a quadratic function is f(x) = a(x - h)² + k, where a ≠ 0 and (h, k ) represents the coordinates of the vertex.
a: Vertical stretch or compression factor.
- a > 0 ⇒ The graph opens upward, and the y-coordinate of the vertex represents the minimum value.
- a < 0 ⇒ The graph opens downward, and the y-coordinate of the vertex represents the maximum value.
h: Horizontal translation.
- h > 0 ⇒ The graph shifts "h" units to the right.
- |h| < 0 ⇒ The graph shifts "h" units to the left.
k: Vertical translation.
- k > 0 ⇒ The graph shifts "k" units upward.
- |k| < 0 ⇒ The graph shifts "k" units downward.
Axis of symmetry:
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 x-coordinate of the vertex, (h, k). Hence, the axis of symmetry is: x = h.
[tex]\boxed{\\\begin{minipage}{5.4cm}\\\indent\quad\sf{\large \underline{Axis\:of\:Symmetry:}}\quad\displaystyle\mathsf{x\:=\:\frac{-b}{2a} \:\:\:}\\\end{minipage}}[/tex]
Find the axis of symmetry:
Step 1: Transform the given quadratic function into its general form, h(x) = ax² + bx + c.
Given: h(x) = - (x + 2)² + 8
[tex]\displaystyle\mathsf{\Rightarrow\:h(x) = - (\:x^2 + 2x + 2x + 4\: ) + 8\: \rightarrow \textsf{[\:Expand the binomial using the \textbf{FOIL method\:}\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\: h(x) = - (\:x^2 + 2x + 2x + 4\: ) + 8\:\rightarrow \textsf{[\:Combine like terms\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\: h(x) = - (\:x^2 + 4x+ 4\: ) + 8\:\rightarrow \textsf{[\:Distribute the negative sign into the parenthesis\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\: h(x) = - \:x^2 - 4x - 4 + 8\:\rightarrow \textsf{[\:Simplify\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\: h(x) = - \:x^2 - 4x + 4\:\rightarrow \textsf{[\:\textbf{General form}, where: a = -1, b = -4, and c = 4\:].}}[/tex]
Step 2: Solve for the axis of symmetry.
Substitute the derived values for a = -1 and b = -4 into the following formula:
[tex]\boxed{\\\begin{minipage}{6cm}\\\indent\qquad\quad\quad\sf{\large \underline{Axis\:of\:Symmetry:}}\\\\\indent\quad\displaystyle\mathsf{x\:=\:\frac{-b}{2a}\:=\:\frac{-(-4)}{2(-1)}\:=\:\frac{4}{-2}\:=\:-2. }\\\end{minipage}}[/tex]
Step 3: Find the maximum value.
Substitute the derived value for the axis of symmetry, x = -2, into h(x) = -x² - 4x + 4.
h(x) = -x² - 4x + 4 ⇒ General form.
[tex]\displaystyle\mathsf{\Rightarrow\:h(-2) = - \:(-2)^2 - 4(-2) + 4\: \rightarrow \textsf{[\:BPEMDAS: Exponent\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow h(-2) = - \:4 - 4(-2) + 4\:\rightarrow \textsf{[\:BPEMDAS: Multiplication\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow\:h(-2) = - 4 + 8 + 4\:\rightarrow \textsf{[\:BPEMDAS: Addition\:].}}[/tex]
[tex]\displaystyle\mathsf{\Rightarrow h(-2) = 8\:\rightarrow \: \textsf{[\:\textbf{Maximum value}\:].}}[/tex]
Hence, the maximum value (vertex) occurs at point (-2, 8).
Graph the parabola:
Find other points to plot:
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.
Solve for the y-intercept:
Solve for the y-intercept by setting x = 0:
Vertex form: h(x) = - (x + 2)² + 8
⇒ h(0) = -(0 + 2)² + 8 → [Let x = 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).
Graph the axis of symmetry:
We can graph the axis of symmetry by drawing a vertical line 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 axis of symmetry, 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).
Final Answer:
The maximum value occurs at point (-2, 8), where x = -2 represents the axis of symmetry.
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