How to Graph Quadratic Functions
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Quadratic Functions
A quadratic has the form \(f(x)=ax^2+bx+c\), where \(a\ne0\). Its graph is a parabola, and its most important ACT features are vertex, axis of symmetry, intercepts, and opening direction.
Core Formulas
Vertex form is \(f(x)=a(x-h)^2+k\), with vertex \((h,k)\) and axis \(x=h\). In standard form, the vertex x-coordinate is \(x=-\frac{b}{2a}\).
ACT Strategy
Use vertex form when it is given, factor to find intercepts when possible, and remember that \(a>0\) opens upward while \(a<0\) opens downward.
Reference Graphs and Visuals
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Graphing Quadratic Functions
Think of this lesson as more than a rule to memorize. Graphing Quadratic Functions is about parabolas, roots, factoring, and graph behavior. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
A quadratic graph has vertex form \(y=a(x-h)^2+k\), where \((h,k)\) is the turning point and \(a\) controls width and direction.
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Read the scale and labels first.
- Identify the key values the graph shows.
- Connect the graph to the formula or data table.
- Answer using the units and context.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Graphing Quadratic Functions
1) \(Find the vertex of y=(x-3)^2+2.\)
2) \(Does y=-2(x+1)^2-4 open up or down?\)
3) \(Find the axis of symmetry of y=(x-5)^2-9.\)
4) \(Find the y-intercept of y=x^2-4x+1.\)
5) \(Find the vertex of y=x^2+6x+8.\)
6) \(Find the x-intercepts of y=(x-2)(x+4).\)
7) \(If f(x)=3x^2+2x-5, find f(-1).\)
8) \(Find the x-intercepts of y=2(x-1)^2-8.\)
9) \(Find the range of y=-3(x+2)^2+7.\)
10) \(Find the minimum value of y=x^2-10x+21.\)
11) \(A parabola has vertex (-2,5) and passes through (0,13). Write its equation in vertex form.\)
12) \(Find the vertex and axis of y=-x^2+8x-10.\)
13) \(Find the x-intercepts of y=2x^2-7x+3.\)
14) \(Which graph is narrower: y=4x^2 or y=\frac12x^2?\)
15) \(For y=(x+3)^2-16, find all intercepts.\)
16) \(The height of a ball is h(t)=-16t^2+48t+5. What is its maximum height?\)
17) \(Describe y=-\frac12(x-4)^2+6 as a transformation of y=x^2.\)
18) \(A quadratic has roots -2 and 6 and y-intercept -24. Find its vertex.\)
19) \(For what value of k does y=x^2-6x+k touch the x-axis exactly once?\)
20) \(For y=-x^2+12x-27, find where the graph is above the x-axis and its maximum value.\)
1)\(Find the vertex of y=(x-3)^2+2.\)
Step 1: Compare with \(y=a(x-h)^2+k\).
Step 2: Here \(h=3\) and \(k=2\).
Answer: \((3,2)\)
2)\(Does y=-2(x+1)^2-4 open up or down?\)
Step 1: Identify \(a=-2\).
Step 2: A negative leading coefficient opens the parabola downward.
Answer: down
3)\(Find the axis of symmetry of y=(x-5)^2-9.\)
Step 1: In vertex form, the axis is \(x=h\).
Step 2: Here \(h=5\).
Answer: \(x=5\)
4)\(Find the y-intercept of y=x^2-4x+1.\)
Step 1: Set \(x=0\).
Step 2: \(y=0^2-4(0)+1=1\).
Answer: \((0,1)\)
5)\(Find the vertex of y=x^2+6x+8.\)
Step 1: \(x=-\frac{b}{2a}=-\frac{6}{2}=-3\).
Step 2: \(y=(-3)^2+6(-3)+8=-1\).
Answer: \((-3,-1)\)
6)\(Find the x-intercepts of y=(x-2)(x+4).\)
Step 1: Set each factor equal to 0.
Step 2: \(x-2=0\) or \(x+4=0\).
Answer: \((2,0)\text{ and }(-4,0)\)
7)\(If f(x)=3x^2+2x-5, find f(-1).\)
Step 1: Substitute \(x=-1\).
Step 2: \(3(-1)^2+2(-1)-5=3-2-5=-4\).
Answer: \(-4\)
8)\(Find the x-intercepts of y=2(x-1)^2-8.\)
Step 1: Set \(2(x-1)^2-8=0\).
Step 2: \((x-1)^2=4\), so \(x-1=\pm2\).
Answer: \((-1,0)\text{ and }(3,0)\)
9)\(Find the range of y=-3(x+2)^2+7.\)
Step 1: The vertex is \((-2,7)\).
Step 2: Since \(a<0\), the vertex gives the maximum.
Answer: \(y\le7\)
10)\(Find the minimum value of y=x^2-10x+21.\)
Step 1: \(x=-\frac{-10}{2}=5\).
Step 2: \(5^2-10(5)+21=-4\).
Answer: \(-4\)
11)\(A parabola has vertex (-2,5) and passes through (0,13). Write its equation in vertex form.\)
Step 1: Use \(y=a(x+2)^2+5\).
Step 2: Substitute \((0,13)\): \(13=4a+5\), so \(a=2\).
Answer: \(y=2(x+2)^2+5\)
12)\(Find the vertex and axis of y=-x^2+8x-10.\)
Step 1: \(x=-\frac{8}{2(-1)}=4\).
Step 2: \(y=-16+32-10=6\).
Answer: \(\text{vertex }(4,6),\ \text{axis }x=4\)
13)\(Find the x-intercepts of y=2x^2-7x+3.\)
Step 1: Factor: \(2x^2-7x+3=(2x-1)(x-3)\).
Step 2: Solve each factor equal to 0.
Answer: \(\left(\frac12,0\right)\text{ and }(3,0)\)
14)\(Which graph is narrower: y=4x^2 or y=\frac12x^2?\)
Step 1: Compare \(|a|\) values.
Step 2: The larger \(|a|\) value makes a narrower graph.
Answer: \(y=4x^2\)
15)\(For y=(x+3)^2-16, find all intercepts.\)
Step 1: Set \(0=(x+3)^2-16\) to get \(x=-7,1\).
Step 2: Set \(x=0\) to get \(y=-7\).
Answer: \((-7,0),\ (1,0),\ (0,-7)\)
16)\(The height of a ball is h(t)=-16t^2+48t+5. What is its maximum height?\)
Step 1: The maximum occurs at \(t=-\frac{48}{2(-16)}=\frac32\).
Step 2: \(h\left(\frac32\right)=-36+72+5=41\).
Answer: \(41\)
17)\(Describe y=-\frac12(x-4)^2+6 as a transformation of y=x^2.\)
Step 1: \(x-4\) shifts right 4 and \(+6\) shifts up 6.
Step 2: \(-\frac12\) reflects over the x-axis and widens the graph.
Answer: right 4, up 6, reflected downward, and wider
18)\(A quadratic has roots -2 and 6 and y-intercept -24. Find its vertex.\)
Step 1: Use \(y=a(x+2)(x-6)\) and \(-24=-12a\), so \(a=2\).
Step 2: The axis is halfway between the roots: \(x=2\); then \(y=2(4)(-4)=-32\).
Answer: \((2,-32)\)
19)\(For what value of k does y=x^2-6x+k touch the x-axis exactly once?\)
Step 1: Touching once means discriminant 0.
Step 2: \((-6)^2-4(1)k=36-4k=0\), so \(k=9\).
Answer: \(9\)
20)\(For y=-x^2+12x-27, find where the graph is above the x-axis and its maximum value.\)
Step 1: Solve \(-x^2+12x-27=0\) to get roots 3 and 9.
Step 2: The graph opens down, so it is above the axis between the roots; the vertex at \(x=6\) has value 9.
Answer: \(3<x<9;\ \text{maximum }9\)
Graphing Quadratic Functions Practice Quiz
